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logarithm/149457: This question is from textbook The instructions say for each function, find f^-1. Help please, step by step. Thank you... f(x)= 6^x-1 Also, solve each equation. Find the Exact solutions. 1/2^x = 5 the exponent is above the 2 only. 1 solutions Answer 109646 by jim_thompson5910(28598)   on 2008-07-23 17:42:16 (Show Source): You can put this solution on YOUR website! Start with the given function. Switch x and f(x). Take the log of both sides. Rewrite the right side using the identity Distribute Add to both sides. Combine the logs using the identity Divide both sides by to isolate f(x) Use the change of base formula to rewrite the left side. So the inverse function is Start with the given equation. Multiply both sides by . Divide both sides by . Take the log of both sides. Rewrite the right side using the identity Divide both sides by to isolate x Use the change of base formula to rewrite the left side. So the answer is which approximates to

Graphs/149454: 1)line l goes through (1,3) and is parallel to the line through (4,3) and (-3,1) FInd the slope of l. the answer I got was 2/7 is this correct? 2)line l goes through the point (5,-2) and is perpendicular to the line through (-2,1)and (-4,-2) Find the slope of l the answer I got was -3/2 is this correct? 3)Write the equation y=1/3x-2 in standard form using only integers and a positive coefficent for x The answer I got was x+3y=6 is this correct? 4) A Line that goes through (-1,-3)and has a slope of 4. My answer is y=4x-3 is this correct? 1 solutions Answer 109645 by jim_thompson5910(28598)   on 2008-07-23 17:29:53 (Show Source): You can put this solution on YOUR website! 1) correct 2) You're on the right track. The slope through (-2,1)and (-4,-2) is 3/2. So flip the fraction and change the sign to get -2/3. So the perpendicular slope is -2/3 3) Close, but you forgot to multiply the term "3y" by negative 1 (since you multiplied the "x" term by -1). So the correct equation is x-3y=6 4) If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Rewrite as Rewrite as Distribute Multiply and to get Subtract 3 from both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line with a slope of which goes through the point (,) is: which is now in form where the slope is and the y-intercept is Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver) Graph of through the point (,) and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.

Polynomials-and-rational-expressions/149449: If someone can please help me with these problems.I have answered them and want to know if my answers are correct if not how can I successfully answer these questions right...... 1)s/6-2/9=0 The answer I came up with is 1/3 is this correct? 2)Solve -43 less than 20 -9v less than or greater to -7 The answer I came up with is (1,7]is this correct? 3)Complete the ordered pair so that it satifies the given equation. 2x-5y=-12:(-1, ) the answer I came up with is (-1,4) is this correct? 4)Find the x and y intercepts of 7x-2y=-5 5)Find the slope of the line that contains the points (-4,-2) and (1,1) 1 solutions Answer 109637 by jim_thompson5910(28598)   on 2008-07-23 16:54:40 (Show Source): You can put this solution on YOUR website! Start with the given equation. Multiply both sides by the LCD 18 to clear out the fractions. Distribute and multiply. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . ---------------------------------------------------------------------- Answer: So the answer is Which approximates to 2) Start with the given compound inequality. Subtract from all sides. Combine like terms on the left side. Combine like terms on the right side. Divide all sides by -9. Note: dividing all sides of the inequality will flip the signs Reduce. Rearrange the inequality. So the answer in interval notation is [) Also, the answer in set-builder notation is Here's the graph of the solution set Graph of the solution set Note: There is a closed circle at which means that we're including this value in the solution set Also, there is an open circle at which means that we're excluding this value from the solution set. 3) Start with the given equation. Plug in . Multiply. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the ordered pair is (-1,2) Note: you can plug in the numbers x=-1 and y=2 into the equation to check your answer. 4) Start with the given equation Let's find the x-intercept To find the x-intercept, let y=0 and solve for x: Plug in Simplify Divide both sides by 7 So the x-intercept is (note: the x-intercept will always have a y-coordinate equal to zero) ------------------ Start with the given equation Now let's find the y-intercept To find the y-intercept, let x=0 and solve for y: Plug in Simplify Divide both sides by -2 Reduce So the y-intercept is (note: the y-intercept will always have a x-coordinate equal to zero) ------------------------------------------ So we have these intercepts: x-intercept: y-intercept: 5) Start with the slope formula. Plug in , , , , , Subtract from to get Subtract from to get So the slope of the line that goes through the points and is

Functions/149447: f(x)=8x2-4x-5 for the given function f match each domain value(-1,0,1/2,2) with its corresponding range value. 1 solutions Answer 109635 by jim_thompson5910(28598)   on 2008-07-23 16:45:05 (Show Source): You can put this solution on YOUR website! With the given domain values are -1,0,1/2, and 2, we can find the corresponding range values by plugging in each value. So I'll do the first one and the third one. # 1 Let's find the corresponding range value for Start with the given equation. Plug in . Square to get . Multiply and to get . Multiply and to get . Combine like terms. So with the given domain value of , the corresponding range value is ----------------- # 3 Let's find the corresponding range value for Start with the given equation. Plug in . Square to get . Multiply and to get . Multiply and to get . Combine like terms. So with the given domain value of , the corresponding range value is

Sequences-and-series/149448: to find the first 3 terms of the Recursively defined sequence: a1=5 , an=2a n-1 for the answer I got 5,10,20... is this correct? 1 solutions Answer 109633 by jim_thompson5910(28598)   on 2008-07-23 16:38:27 (Show Source): You can put this solution on YOUR website! Since , this means that... when n=2, then --------- when n=3, then ---------------- So the sequence is 5,10,20. So you are correct.

Exponential-and-logarithmic-functions/149444: Please help me figure this out - step-by-step. 4y^-1/2 = 16 Thanks. 1 solutions Answer 109630 by jim_thompson5910(28598)   on 2008-07-23 16:34:16 (Show Source): You can put this solution on YOUR website! Start with the given equation. Rewrite as Multiply both sides by Divide both sides by 16. Reduce. Rearrange the equation. Convert from rational notation to radical notation. Square both sides. Square to get So the answer is

Trigonometry-basics/149438: Please help me finde the value of cos^-1(0) Thank you, Natalie 1 solutions Answer 109628 by jim_thompson5910(28598)   on 2008-07-23 16:24:16 (Show Source): You can put this solution on YOUR website! What is asking for is an angle. So this means that . So, let's reference the unit circle From the picture, we can see that the point (0,1) tells us that or (remember, the x coordinate corresponds to cosine). Start with the given equation. Take the inverse cosine of both sides. This will eliminate the "cos" on the left side. So (in degrees) or (in radians)

Trigonometry-basics/149423: Find the exact value of: sin 70degrees cos 40degrees - cos 70degrees sin 40degrees. 1 solutions Answer 109624 by jim_thompson5910(28598)   on 2008-07-23 15:30:24 (Show Source): You can put this solution on YOUR website! Start with the given expression. Rewrite the expression using the identity Subtract Take the sine of 30 to get 1/2 So

Sequences-and-series/149421: find the sum of the geometric series 6 [SUM] 2^n n=1 n=1 2^1=2 n=2 2^2=4 n=3 2^3=8 n=4 2^4=16 n=5 2^5=32 n=6 2^6=64 2+4+8+16+32+64= 126 am I doing this correctly? 1 solutions Answer 109623 by jim_thompson5910(28598)   on 2008-07-23 15:19:40 (Show Source): You can put this solution on YOUR website! You got it. Good job.

Sequences-and-series/149402: how do you figure out the nth term of : 21,19,17,15... please help!!! xxx 1 solutions Answer 109615 by jim_thompson5910(28598)   on 2008-07-23 14:30:19 (Show Source): You can put this solution on YOUR website! Lets assume this sequence is an arithmetic sequence. The general form of the arithmetic sequence is where is the nth term, d is the difference, and is the first term So lets find the difference between 2 terms (i.e. the difference between two terms) ==================================================================================================================== To find the difference, simply pick any term and subtract the previous term from that selected term Select the 2nd term (which is 19) and subtract the 1st term (which is 21) from it. So we get a difference of Lets pick another pair of terms to verify: Select the 3rd term (which is 17) and subtract the 2nd term (which is 19) from it. And again, we get a difference of ----------------------------------------------------- Lets pick another pair of terms to verify: Select the 4th term (which is 15) and subtract the 3rd term (which is 17) from it. And again, we get a difference of ----------------------------------------------------- ==================================================================================================================== Since we've tested every consecutive pair of terms, we've verified that the sequence has a constant difference of . This means the sequence is arithmetic Since the difference is and the first term is , this means the arithmetic sequence is where starts at Check: Notice if we plug in we get plug in Multiply Add which is our first term Notice if we plug in we get plug in Multiply Add which is our second term Notice if we plug in we get plug in Multiply Add which is our third term Notice if we plug in we get plug in Multiply Add which is our fourth term Since each term corresponds to the terms of the given list, this verifies our sequence. ------------------------------------------------------------------------------------------------------------------------ Answer: So the list of numbers 21,19,17,15... can be generated by the sequence where starts at

logarithm/149398: Solve the exponential or logarithmic equations. (Write your final answer in BOTH exact and approximate value). (a) 6^x-3 = 2^x (b) log(5x-6) = 2log x 1 solutions Answer 109613 by jim_thompson5910(28598)   on 2008-07-23 14:25:14 (Show Source): You can put this solution on YOUR website! I'll do the first one to get you started. a) Start with the given equation. Take the log of both sides. Rewrite both sides using the identity Distribute. Subtract from both sides. Add to both sides. Factor out the GCF "x" Combine the logs using the identity Divide. Divide both sides by to isolate x Rewrite the expression using the identity Raise 6 to the 3rd power to get 216 Use the change of base formula to rewrite the right side. So the exact answer is which approximates to

logarithm/149400: Rewrite each expression in exponential form and determine the value of x. (a) log49 x=1/2 (b) log9 27 = x 1 solutions Answer 109609 by jim_thompson5910(28598)   on 2008-07-23 14:16:03 (Show Source): You can put this solution on YOUR website! a) Start with the given equation. Rewrite the equation using the property: ====> Convert from rational notation to radical notation. Take the square root of 49 to get 7. Note: only the positive square root is considered in this case. So the answer is b) Start with the given equation. Rewrite the equation using the property: ====> Rewrite as and as Multiply the exponents. Since the bases are equal, the exponents are equal. Divide both sides by 2. So the answer is

Quadratic_Equations/149395: What type of solution do you get for quadratic equations where D<0? Give reasons for your answer. Also provide an example of such a quadratic equation and find the solution of the equation. 1 solutions Answer 109606 by jim_thompson5910(28598)   on 2008-07-23 14:09:40 (Show Source): You can put this solution on YOUR website! If , then the quadratic will have two complex (ie non real) solutions. For example, let's find the discriminant for From we can see that , , and Start with the discriminant formula Plug in , , and Square to get Multiply to get Subtract from to get Since the discriminant is less than zero, this means that there are two complex solutions Now let's use the quadratic formula to find the solutions of Start with the quadratic formula Plug in , , and Square to get . Multiply to get Subtract from to get Multiply and to get . Take the square root of to get . or Break up the expression. or Break up the fraction for each case. or Reduce. So our answers are or Since our answers are complex, this verifies our original claim.

Quadratic_Equations/149392: Please help with the following word problem: A stone is thrown upward from a bridge. The stone's height in feet, "h", above the water "t" seconds after the stone is thrown is a function given by h=-16t2 + 32t + 256. a. How tall is the bridge from which the stone was thrown? My answer is 256 ft. b. How high above the water is the stone after 3 seconds? My answer is 208 ft. c. What is the maximum height of the stone? Stuck on this question. d. After how many seconds does the stone reach the maximum height? Stuck on this question too. e. How long does it take for the stone to hit the water? My answer is 5.12 seconds. Please help with questions c and d. Any help would be greatly appreciated. Thank you. 1 solutions Answer 109604 by jim_thompson5910(28598)   on 2008-07-23 14:05:47 (Show Source): You can put this solution on YOUR website! a) correct b) correct c) To find the max height, we need to find the vertex. In order to find the vertex, we first need to find the x-coordinate of the vertex. To find the x-coordinate of the vertex, use this formula: . Start with the given formula. From , we can see that , , and . Plug in and . Multiply 2 and to get . Divide. So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also . Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex. Start with the given equation. Plug in . Start with the given equation. Plug in . Square to get . Multiply and to get . Multiply and to get . Combine like terms. So the y-coordinate of the vertex is . So the vertex is . So the highest point on the graph of is . This means that maximum height of the stone is 272 feet From the previous solution, the highest point is at the point . So when t=1, then h=272. So the stone reaches the highest point at 1 second e) correct.

Angles/149389: the measure of angle j is half of its compliment plus 12 degrees 1 solutions Answer 109602 by jim_thompson5910(28598)   on 2008-07-23 13:57:51 (Show Source): You can put this solution on YOUR website! "measure of angle j is half of its compliment plus 12 degrees" translates to Also, since angle j and angle c are complementary angles, this means that Plug in Multiply every term by 2 to clear the fraction. Subtract 12 from both sides. Combine like terms. Divide both sides by 3. Go back to the first equation. Plug in Add Multiply.

Permutations/149387: From a club of 25 members, how many ways can a president, vice-president, secretary, and treasurer be selected? which formula is used and does n=25 and r=4? 1 solutions Answer 109600 by jim_thompson5910(28598)   on 2008-07-23 13:41:48 (Show Source): You can put this solution on YOUR website! Since order does matter, we must use the permutation formula: Start with the given formula Plug in and Subtract to get 21 Expand 25! Expand 21! Cancel like terms. Simplify Now multiply 25*24*23*22 to get 303,600 So 25 choose 4 (where order does matter) yields 303,600 unique combinations

Miscellaneous_Word_Problems/149388: Find the multiplicative inverse of the following number: -0.25 1 solutions Answer 109598 by jim_thompson5910(28598)   on 2008-07-23 13:37:16 (Show Source): You can put this solution on YOUR website! First write -0.25 as the fraction Now to find the multiplicative inverse, simply flip the fraction to get or So the multiplicative inverse of -0.25 is -4 Check: Simply multiply the two numbers -0.25 and -4 to get Since their product is 1, this means that -4 is the multiplicative inverse of -0.25 (and vice versa).

Linear-equations/149380: graph the line and find the slope (-2,5) (3,0) find the slope of the line m= 1 solutions Answer 109587 by jim_thompson5910(28598)   on 2008-07-23 11:59:15 (Show Source): You can put this solution on YOUR website! First graph the points (-2,5) and (3,0) Now draw a line through those points Now start at the point (-2,5) and move 5 units down (to get to the same level as the second point). Since we went 5 units down, this means that the rise is -5. Now move 5 units to the right to get to the next point. Since we went 5 units to the right, this means that the run is 5. So the rise is -5 and the run is 5. This makes the slope So the slope between the two points (-2,5) and (3,0) is

Inequalities/149361: 3-3x>-6 1 solutions Answer 109586 by jim_thompson5910(28598)   on 2008-07-23 11:46:28 (Show Source): You can put this solution on YOUR website! Start with the given inequality. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . note: Remember, the inequality sign flips when we divide both sides by a negative number. Reduce. ---------------------------------------------------------------------- Answer: So the answer is So the answer in interval notation is Also, the answer in set-builder notation is Here's the graph of the solution set

Linear-equations/149376: 4x + 3y = 12 1 solutions Answer 109585 by jim_thompson5910(28598)   on 2008-07-23 11:43:33 (Show Source): You can put this solution on YOUR website! Do you want to graph? Start with the given equation. Subtract from both sides. Divide both sides by to isolate . Simplify. Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is -4 and the run is 3. This means that to go from point to point, we can go down 4 and over 3 So starting at , go down 4 units and to the right 3 units to get to the next point Now draw a line through these points to graph So this is the graph of through the points and

Rational-functions/149317: a^2b/b^2a * ab-b^2/ab-a^2 1 solutions Answer 109555 by jim_thompson5910(28598)   on 2008-07-22 23:31:12 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get . Factor to get . Combine the fractions. Multiply. Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to .

Rational-functions/149318: w-4/3w ÷ 2w-8/9w 1 solutions Answer 109554 by jim_thompson5910(28598)   on 2008-07-22 23:18:11 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply the first fraction by the reciprocal of the second fraction . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to .

Linear-equations/149331: Solve the following: Passing through (5,-9) and perpendicular to x + 7y = 12 1 solutions Answer 109553 by jim_thompson5910(28598)   on 2008-07-22 23:11:17 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract x from both sides. Divide both sides by 7. Break up the fraction and simplify. We can see that the equation has a slope and a y-intercept . Now to find the slope of the perpendicular line, simply flip the slope to get . Now change the sign to get . So the perpendicular slope is . Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope and the coordinates of the given point . Start with the point slope formula Plug in , , and Rewrite as Distribute Multiply Subtract 9 from both sides. Combine like terms. So the equation of the line perpendicular to that goes through the point is . Here's a graph to visually verify our answer: Graph of the original equation (red) and the perpendicular line (green) through the point .

Exponential-and-logarithmic-functions/149307: This question is from textbook College Algebra How can I solve this problem if I do not have a value for r? In all the examples I see, there is always a value given for r. 5(1+(r/360)^720 = 12 1 solutions Answer 109535 by jim_thompson5910(28598)   on 2008-07-22 20:22:32 (Show Source): You can put this solution on YOUR website! Is your equation ??? You're missing a parenthesis. Start with the given equation. Divide both sides by 5. Subtract 1 from both sides. Combine like terms. Take the 720th root of both sides. Multiply both sides by 360.

Equations/149304: 5t^-16t=-12 i have tried everything. please help me. I am so greatful for this site and your time. thank you so much. This is factoring equations. 1 solutions Answer 109532 by jim_thompson5910(28598)   on 2008-07-22 20:06:26 (Show Source): You can put this solution on YOUR website! Start with the given equation. Add 12 to both sides. Now let's factor the left side Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is . Now multiply the first coefficient by the last term to get . Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ? To find these two numbers, we need to list all of the factors of (the previous product). Factors of : 1,2,3,4,5,6,10,12,15,20,30,60 -1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*60 2*30 3*20 4*15 5*12 6*10 (-1)*(-60) (-2)*(-30) (-3)*(-20) (-4)*(-15) (-5)*(-12) (-6)*(-10) Now let's add up each pair of factors to see if one pair adds to the middle coefficient : First Number Second Number Sum 1 60 1+60=61 2 30 2+30=32 3 20 3+20=23 4 15 4+15=19 5 12 5+12=17 6 10 6+10=16 -1 -60 -1+(-60)=-61 -2 -30 -2+(-30)=-32 -3 -20 -3+(-20)=-23 -4 -15 -4+(-15)=-19 -5 -12 -5+(-12)=-17 -6 -10 -6+(-10)=-16 From the table, we can see that the two numbers and add to (the middle coefficient). So the two numbers and both multiply to and add to Now replace the middle term with . Remember, and add to . So this shows us that . Replace the second term with . Group the terms into two pairs. Factor out the GCF from the first group. Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis. Combine like terms. Or factor out the common term So factors to . Note: you can check to see if you did it right by FOILing to get . So becomes . Now set each factor equal to zero: or or Now solve for t in each case So our answers are or

Linear-systems/149290: If the circumference of a circle is 12 meters, then what is its diameter? 1 solutions Answer 109513 by jim_thompson5910(28598)   on 2008-07-22 18:23:02 (Show Source): You can put this solution on YOUR website! Start with the circumference of a circle formula. Plug in Divide both sides by Simplify. So the radius is 6 meters and the diameter is 12 meters (simply double the radius to get the diameter).

Graphs/149240: Solve the inequality 1 solutions Answer 109486 by jim_thompson5910(28598)   on 2008-07-22 14:13:00 (Show Source): You can put this solution on YOUR website! First, we need to find the vertical asymptote(s) Vertical Asymptote: To find the vertical asymptote, just set the denominator equal to zero and solve for x Set the denominator equal to zero Add 7 to both sides Combine like terms on the right side So the vertical asymptote is -------------------- Now we need to find any x-intercepts Start with the given equation Plug in Multiply both sides by . Add 4 to both sides. So the x-intercept is (4,0) --------------- This means that we'll have to test three regions Region 1: This region is from negative infinity to the x-intercept So let's test the value Start with the given equation Plug in Simplify. Since is greater than or equal to zero, this means that every point in the interval (] is above the x-axis. So the interval (] is part of the solution to the inequality ----------- Region 2: This region is from the x-intercept to the vertical asymptote So let's test the value Start with the given equation Plug in Simplify. Since is not greater than or equal to zero, this means that every point in the interval () is below the x-axis. So the interval () is not part of the solution to the inequality ----------- Region 3: This region is from the vertical asymptote to positive infinity So let's test the value Start with the given equation Plug in Simplify. Since is greater than or equal to zero, this means that every point in the interval () is above the x-axis. So the interval () is part of the solution to the inequality So that means that the solution is (]()

Graphs/149237: Given , A) Find the domain. B) Determine the vertical asymptote(s). C) Determine the horizontal asymptote or oblique asymptote. D) Find the y-intercept. E) Find the x-intercept(s). 1 solutions Answer 109485 by jim_thompson5910(28598)   on 2008-07-22 13:54:55 (Show Source): You can put this solution on YOUR website! A) Domain: Start with the given function Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain. Factor the left side (note: if you need help with factoring, check out this solver) Now set each factor equal to zero: or or Now solve for x in each case So our solutions are or Since and make the denominator equal to zero, this means we must exclude and from our domain So our domain is: which in plain English reads: x is the set of all real numbers except or So our domain looks like this in interval notation note: remember, the parenthesis excludes -5 and 5 from the domain -------------------------------------------------- B) Vertical Asymptote: To find the vertical asymptote, just set the denominator equal to zero and solve for x Set the denominator equal to zero Add 25 to both sides Combine like terms on the right side Take the square root of both sides or Simplify So the vertical asymptotes are or -------------------------------------------------- Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is . C) Horizontal/Oblique Asymptote: Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure: To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator. Looking at the numerator , the leading coefficient is Looking at the denominator , the leading coefficient is So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide by to get So the horizontal asymptote is -------------------------------------------------- D) Y-Intercept: To find the y-intercept, simply plug in Start with the given function Plug in Simplify So the y-intercept is -------------------------------------------------- E) X-Intercept(s): To find the x-intercept(s), simply plug in and solve for x Start with the given function Plug in Since the denominator cannot be equal to zero, this means that the numerator is equal to zero. Add 18 to both sides. Divide both sides by 2. or Take the square root of both sides. So the x-intercepts are and ------------------------------ Notice if we graph , we can visually verify our answers: Graph of with the horizontal asymptote (blue line) and the vertical asymptotes and (green lines)

Graphs/149236: Graph 1 solutions Answer 109484 by jim_thompson5910(28598)   on 2008-07-22 13:52:02 (Show Source): You can put this solution on YOUR website! First, let's find the asymptotes of the equation Horizontal Asymptote: Since the degree of the numerator (which is ) is less than the degree of the denominator (which is ), the horizontal asymptote is always So the horizontal asymptote is -------------------------------------------------- Vertical Asymptote: To find the vertical asymptote, just set the denominator equal to zero and solve for x Set the denominator equal to zero Add 4 to both sides Combine like terms on the right side Take the square root of both sides or Break up the expression and simplify. So the vertical asymptotes are or ------------------ Now we need to test each region to see if it lies above or below the x-axis Region 1: This region is to the left of the vertical asymptote So let's plug in Start with the given equation. Plug in . Simplify. Since the y-value is negative, this means that every point in the interval is below the x-axis. ------ Region 2: This region lies between the vertical asymptote and the x-axis So let's plug in Start with the given equation. Plug in . Simplify. Since the y-value is positive, this means that every point in the interval is above the x-axis. ------ Region 3: This region lies between the x-axis and the vertical asymptote So let's plug in Start with the given equation. Plug in . Simplify. Since the y-value is negative, this means that every point in the interval is below the x-axis. ----------------- Region 3: This region lies to the right of the vertical asymptote So let's plug in Start with the given equation. Plug in . Simplify. Since the y-value is positive, this means that every point in the interval is above the x-axis. ---------------- So with all of this information, we can now graph the function

Graphs/149234: Solve the inequality (x + 3)(x + 1)(x – 5) < 0 and write the solution set both in interval notation and in set notation. Show work/explanation. 1 solutions Answer 109482 by jim_thompson5910(28598)   on 2008-07-22 13:36:45 (Show Source): You can put this solution on YOUR website! Start with the given inequality Set the left side equal to zero Set each individual factor equal to zero: , or Solve for x in each case: , or So our critical values are , and Now set up a number line and plot the critical values on the number line So let's pick some test points that are near the critical values and evaluate them. Let's pick a test value that is less than (notice how it's to the left of the leftmost endpoint): So let's pick Start with the given inequality Plug in Evaluate and simplify the left side Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set. So part our solution in interval notation is () --------------------------------------------------------------------------------------------- Let's pick a test value that is in between and : So let's pick Start with the given inequality Plug in Evaluate and simplify the left side Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it. --------------------------------------------------------------------------------------------- Let's pick a test value that is in between and : So let's pick Start with the given inequality Plug in Evaluate and simplify the left side Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set. So part our solution in interval notation is () --------------------------------------------------------------------------------------------- Let's pick a test value that is greater than (notice how it's to the right of the rightmost endpoint): So let's pick Start with the given inequality Plug in Evaluate and simplify the left side Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it. --------------------------------------------------------------------------------------------- Summary: So the solution in interval notation is: () () Also, the answer in set notation is Here's a graph to visually verify our answer: Graph of

Geometric_formulas/149173: 10. The segments GA and GB are tangent to a circle with center O at A and B, and AGB is a 60-degree angle. Given that GA = 12 square root 3 cm, find the distance GO. Find the distance from G to the nearest point on the circle. 1 solutions Answer 109434 by jim_thompson5910(28598)   on 2008-07-21 23:51:37 (Show Source): You can put this solution on YOUR website! First let's draw the picture (take note that I'm cutting AGB in half to make 2 30 degree angles): From the drawing, we can see that segments GA and AO are legs of a triangle while GO is the hypotenuse. Since the angle that we are working with is the 30 degree angle, this means that GA is the adjacent side. So to find the hypotenuse, we need to use the cosine function: Start with the cosine function. Plug in the given sides and angle. Multiply both sides by Divide both sides by Evaluate to get Multiply by the reciprocal Cancel like terms. Simplify Multiply So the length of GO is 24 units. The closest point to G will lie on the line GO. So to find the distance from G to this point, we need to find the radius. Using the previous drawing, we can see that the radius is AO. To find the length of AO, note that AO is the opposite leg to the angle 30 degrees. So let's use the sine function to find AO Start with the sine function. Plug in the angle 30 , the opposite side AO, and the hypotenuse 24 Take the sine of 30 to get Cross multiply. Divide both sides by 2. So the length of AO (the radius) is 12 units. Now simply subtract the radius from the length GO to get So the distance from G to the nearest point on the circle is 12 units.