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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.
 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, Natalie1 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 x1 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
 Angles/149389: the measure of angle j is half of its compliment plus 12 degrees1 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 = 121 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^21 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-8/9w1 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 = 121 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 = 121 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!

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 NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+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).