Questions on Algebra: Functions, Domain, NOT graphing answered by real tutors!

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a junior College awarded 26 varsity letters in crew, 15 in swimming, and 16 in soccer. If awards went to 46 students and only 2 lettered in all sports how many students lettered in 2 of the 3 sports?
:
Eliminate the 2 that lettered in all 3 sport
46 - 2 = 44 students
:
26 - 2 = 24 crew
15 - 2 = 13 swim
16 - 2 = 14 soccer
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total = 51 awards given to 44 students
:
51 - 44 = 7 students lettered in two sports
:
Check 7(2) + 37(1) = 51

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Without repeats:
7!=7*6*5*4*3*2*1=5,040 ans.

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Which of the following sets are functions from the set of the first components to the set of second components?
• { (b,a), (d,c), (a,e), (g,f) } this is a function
• { (a,b), (b,a), (c,c), (a,c) } not a function because "a" has two y-values
• { (b,a), (c,a), (b,b), (c,b) } not a function because "b" has two y-values
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Cheers,
Stan H.

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any given x value cannot have 2 different y values
1st set appears to be a function
2nd set appears to be a function
3rd set is not a function as b has two different values a and b

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the domain and range of each function: ex.
#36 y=absolute value of x+4
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y = |x+4|
Domain: All Real Numbers
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Range: ?
The least value y can have is zero:
So, range is all Real numbers >= zero
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Cheers,
Stan H.

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Domain is D={-6,-3,1,2,3}
Range is R={-4,-1,2,4,7}

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I have to fine the x and y intercept of f(x)=4x-1
I know that by setting f(0) I can get the y intercept (-1) but I am not sure how to find the x intercept. Can someone please explain how to get it. Thanks.
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y = 4x-1
To find the y-intercept, let x = 0; Then y = -1
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To find the x-intercept, let y = 0; Then 0= 4x-1 and x = 1/4
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Cheers,
Stan H.

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We'll see shortly:
At points (-1,-4), and consider other points (0,0):

, "m"
Thru points (-1,-4):
, slope intercept form

, y-intercept=0:
Then, the line eqn follows:
-->(y)=4x ---> points (-1,-4)
If (y)=(1/2)(x), then,
--->(y)=(1/2)x ----> points (-1,-1/2), ANSWER
Thank you,
Jojo

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f(x)= f(x + delta(x)) - f(x)/ delta(x) how was this function derived?
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Look at m(x) = [f(x+delta(x)) - f(x)] / delta(x)
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This is a slope form where the numerator is a change in y and
the denominator is a corresponding change in x.
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It would be the slope of a line between the two points,
(0,f(x)) and (delta(x),f(x+delta(x))
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The form leads to a definition of "the limit of a function" as
delta(x) approaches zero.
================================
Cheers,
Stan H.

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This equation is linear Take 4 values of x and enter the y values.
y=x+1
xvalue y value
0 1
1 2
2 3

3 4
4............

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i will not say i am 100% sure but i am around 90% sure, i believe that the answer is 2/7.
-If you plug in -7/2 for x you will get f(-7/2)=-1/(-7/2)
-If you have a fraction in the denominator you multiply the fraction by its reciprocal. So it would be (-1/1)(2/-7), you will get (-2/-7) also equal to 2/7

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First find (the inner function)


Start with the second function.


Plug in .


Cube to get .


Subtract.



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Since , this means that (just replace with -7)

So let's evaluate


Start with the first function.


Plug in .


Square to get .


Subtract.


Since , this means that


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Answer:


So the solution is



Note: there is another way to do this which involves substituting into and evaluating the composite function at . However, this method is a bit more complicated.

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Start with the first function


Plug in


Remove the parenthesis


Combine like terms.

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Given f(x) = x^2 - 5 and g(x) = sqrt{x}, find h(x) = f(g(x)).
We plug the value of g(x) into f(x) and simplify, if possible.
By doing this, we will find h(x).
f(sqrt{x}) = (sqrt{x})^2 - 5
f(sqrt{x}) = x - 5
So, then h(x) = x - 5
Did you follow?

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To solve, replace x with each integer given in the brackets and simplify.
You will need to do this for each number.
Keep in mind that x = domain and y = range.
So, when you are solving for y, you are actually solving for the range.
Can you take it from here?

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yes

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d varies inversely as w and directly as the square of v.
This translates to:
d =
:
If d=40 when w=6 and v=2,
40 =
Find k
= 40
:
Multiply both sides by 6
4k = 6*40
4k = 240
k =
k = 60
;
The equation: d =


find d when w=9 and v=4.
d =
d =
d =
d = 106

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Yes, you are correct:
Area =

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so yx=k therefore 1.5(8)=12 so 12 is our constant k
now if you have x=20 and want to find y --->you must multiply x time y and make it equal to our constant. hence 20(y)=12(our constant)


another way of doing this is to set up a proportion of
an inverted relationship proportion

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State the domain in set-builder notation and interval notation
H(x)=3(x-2)^4/3
where (x-2)^4/3 is written INSIDE a radical
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Are you multiplying by 3 or do you have the cube root of (x-2)^4/3 ?
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Cheers,
Stan H.

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when x=-3 y=-3
when x=-2 y=-1
when x=-1 y=1
when x=1 y=5
when x=2 y=7
when x=3 y=9

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if the graph is tangent to x-axis then discriminant is 0
then
then




then or
then (positive value)
Answer:

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f(x) = 2x-5/x-3
g( x) = 3x - 2 / 1 -x
fog = f ( g(x) )
= f ( 3x -2/ 1-x)

= [2 (3x-2/1-x) -5]/ (3x-2/1-x)-3
= [ 2 (3x - 2 ) - 5 (1 -x)]/ (1-x) /[(3x - 2 ) - 3 ( 1-x )] / 1-x

= (6x - 4 - 5 +5x) /( 3x - 2 - 3 +3x)
= ( 11 x - 9 ) / (6x -5)

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26 is a relation
28 is a function
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Cheers,
Stan H.

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|AB| = |BF| = R(large circle radius), |CD| = r(small circle radius) The belt section AD and FG are tangent to the large pulley at A and F respectively, so angle DAB as well as GFB are right angles. The line DE is constructed parallel to CB so DE=BC.
we know that EA=6 and ED=12(hypothenuse)
Since triangle AED is a right triangle
we have
AD=FG therefore FG=10.39 inches

Now since ED and BC are parallel, angle AED and angle ABC are equal. Angle AED is the inverse tangent of |AD|/|AE| (opposite/adjacent). that would be equal to arctan of 1.7333 which equals 63.88degrees. The measure of angle ABF is twice the measure of angle ABC and hence you can now find the length of the longer of the two arcs joining A and F. since the angle we are looking for is ABF which = 2(63.88)=127.76. given the radius of 9 and angle of 127.76 we arrive at the small arc distance FA=20.07 inches since the entire circumference equals =56.52. we subtract small arc FA from the entire circumference to get large arc FA. 56.52-20.07 equals
Angle BCD is 180 degrees minus angle ABC and hence you can also find the length of the appropriate arc on the smaller pulley. 180-63.88= 116.12degrees double that and subtract from 360 degrees to get small circle arc DG(smaller portion)
360-232.24= 127.76....how about that!!!. Now given arc angle of 127.76 and a radius of 3 we arrive at arc length of
thusly, adding up all the lengths we have =

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Let C = Chanda's age now
Let S = Suzanne's age now
:
Write an equation for each statement:
:
"Chanda is now four times old as her sister Suzanne."
C = 4S
;
"Next year, Chanda's age will be three times Suzanne's age then."
C + 1 = 3(S + 1)
:
C + 1 = 3S + 3
:
C = 3S + 3 - 1
:
C = 3S + 2
:
How old is each now?
:
From the 1st equation substitute 4S for C in the above equation
4S = 3S + 2
:
4S - 3S = 2
:
S = 2 yrs is Suzanne's age now
:
You figure out C's age

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Multiply both sides by 9 first.
Distribute 9 on the left hand side.
Add 72 to both sides.
Simplify.
Divide by 9.
.
.
.
Here's the two lines graphed and point (6,8) shown.

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It would help if you also shared the directions or the entire question.
Where did you get this question?
What is the name of the chapter and book where you found it?
What exactly do you us to do with this question?
I'll wait for your reply.

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yes and no. (t-5) is greater than or equal to 0. So t is greater than or equal to 5.
This may have been what you meant but I wanted to make sure you made that connection.
.
If you need any more help with logs just email me. justin.sheppard.tech@hotmail.com
MRperkins

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Remember, the domain is simply the set of all x values that produce a y value.

# 1



Start with the given function


Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of "t" 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 t 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: t is the set of all real numbers except t CANNOT equal or t CANNOT equal

So our domain looks like this in interval notation


note: remember, the parenthesis excludes -5 and 5 from the domain



If we wanted to graph the domain on a number line, we would get:

Graph of the domain in blue and the excluded values represented by open circles

Notice we have a continuous line until we get to the holes at and (which is represented by the open circles).
This graphically represents our domain in which t can be any number except t cannot equal -5 or 5

---

# 2





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.



Divide both sides by 4 to isolate x



Divide





Since makes the denominator equal to zero, this means we must exclude from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except x CANNOT equal

So our domain looks like this in interval notation


note: remember, the parenthesis excludes 0 from the domain

If we wanted to graph the domain on a number line, we would get:

Graph of the domain in blue and the excluded value represented by open circle

Notice we have a continuous line until we get to the hole at (which is represented by the open circle).
This graphically represents our domain in which x can be any number except x cannot equal 0

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Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients


So let's list the factors of -3 (the last coefficient):



Now let's list the factors of 1 (the first coefficient):



Now let's divide each factor of the last coefficient by each factor of the first coefficient









Now simplify

These are all the distinct rational zeros of the function that could occur

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In order to write in vertex form, we need to complete the square for


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Completing the Square:


Start with the given expression.


Take half of the coefficient to get . In other words, .


Now square to get . In other words,


Now add and subtract . Notice how . So the expression is not changed.


Group the first three terms.


Factor to get .


So after completing the square, transforms to . So .



So is equivalent to .


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Start with the previous equation


Rewrite as


Place a "1" outside the parenthesis.


Now is in vertex form where , and

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a)(5,infinity) log can't be <=0
b) all real numbers
c) (-4,infinity) same as a).
d) all real numbers.
.
Ed

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A log cannot be negative or zero.
The domain is (5,infinity)
.
Ed

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Let's evaluate the left endpoint


Start with the given equation.


Plug in .


Cube to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


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Let's evaluate the right endpoint


Start with the given equation.


Plug in .


Cube to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So as x changes from -2 to -1, f(x) (or y) changes from -23 to 2 which means that the graph MUST have crossed over the x-axis somewhere in between x=-2 and x=-1. So this shows that there is a zero between x=-2 and x=-1

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you have a vertical line at x=15 y can be any value. so there is no solving to do here...not sure what your trying to find with this I would imagine your suppose to graph this but who knows....good luck

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I am not sure what you need to do here, I am assuming you need to graph?
-4y-8=12x
First put it in slope-intercept form;
-4y=12x+8divide each side by -4;
y=-3x-2
now we have it in slope-intercept form, we graph;

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problem 1:
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$3.00 per pound = ($3.00 / 16) per ounce = 18.75 cents per ounce.
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problem 2:
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5 centimeters per minute = x meters per week.
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first convert 1 week to minutes.
1 week = 7 days * 24 hours per day * 60 minutes per hour = 10080 minutes.
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since your problem states 5 centimeters per minute, and you want to know how many meters per week, you first want to get how many centimeters per week.
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since 1 week is 10080 minutes, and your original equation is for 1 minute, you need to multiply your equation by 10080 to find out how many centimeters in a week.
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5 centimeters per minute = 5 * 10080 centimeters per week = 50400 centimeters per week.
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now you want to convert from centimeters to meters.
100 centimeters = 1 meter, so 1 centimeter = .01 meters (1/100 of a centimeter).
to convert from centimeters to meters you have to divide the number of centimeters by 100.
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50400 centimeters / 100 = 504 meters.
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your answer is: 504 meters per week.
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You are correct. The answer is . If you aren't sure, try plugging in some numbers. For instance, let , and . So

---->

Now let's say that , and but now we do NOT know the value of "t". So this means that

----> ----> ----> (which was the original value we gave to "t")


This isn't a foolproof way to confirm your answer, but it may help you see what's happening.

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Answer: C(x) = 35x + 40
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Cheers,
Stan H.

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B=2a
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That's interesting. Do you have a question?

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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.



Add 5 to both sides


Combine like terms on the right side





Since makes the denominator equal to zero, this means we must exclude from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except x CANNOT equal

So our domain looks like this in interval notation


note: remember, the parenthesis excludes 5 from the domain

If we wanted to graph the domain on a number line, we would get:

Graph of the domain in blue and the excluded value represented by open circle

Notice we have a continuous line until we get to the hole at (which is represented by the open circle).
This graphically represents our domain in which x can be any number except x cannot equal 5

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Factoring the term on the left:

You can cancel like-numbers if found on both numerator and denominator:

Factoring the term on the right:

You can cancel like-numbers if found on both numerator and denominator:

Combining:

You can cancel like-numbers if found on both numerator and denominator:

Factoring numerator:

You can cancel like-numbers if found on both numerator and denominator: