Function Notation - Algebra 2

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Question

Solve for when $f(x)= \frac{x^{2}+12x}{3x}$ .

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Answer

Plug 3 in for x:

$f(3)=\frac{(3^{2})+123}{33}$

Simplify:

= $\frac{9+36}{9}$

= 5

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Question

What is

$(f\cdot g)(x)$ ?

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Answer

To find the composition of two functions, substitute the second equation in to the first function.

$(g\cdot h)(x)=g(h(x))$

Therefore,

$(f\cdot g)(x) = (-8x+8)(x^2)$

and

Thus,

$(f\cdot g)(x)= -8x^3+8x^2$ .

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Question

Solve the function for . When

What does equal when,

$f(x)=x^{2}-9$

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Answer

Plug 16 in for .

Add 9 to both sides.

Take the square root of both sides. $\sqrt{25}$ = $\sqrt{x^{2}}$

Final answer is $\pm5$

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Question

What is of the following equation?

$f(x)=x^{2}+15x+35$

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Answer

To complete an equation with a function, plug the number inside the parentheses into the equation for and solve algebraically.

In this case the $f(7)=7^{2}+(15*7)+35$

Square the 7 and multiply to get

Add the numbers to get the answer .

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Question

Given $f(x) = -2x^{2} - 3x +5$ , find .

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Answer

Plug in a for x:

$f(a)=-(-2a^{2}-3a+5) = 2a^2+3a-5$

Next plug in (a + h) for x:

$f(a+h) = -2(a+h)^2-3(a+h)+5 = -2a^{2} - 4ah - 2h^{2} -3a - 3h +5$

Therefore f(a+h) - f(a) = $-2h^{2} - 4ah - 3h$ .

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Question

$\textup{If }f(p)= p^{3}-3p,;\textup{what is the value of }f(5)\textup{?}$

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Answer

$\textup{Just plug in 5 every time you see }p\textup{ in the equation and solve:}$

$\left ( 5\right )^{3}-3\left ( 5\right )=125-15=110$

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Question

$\textup{If }f\left(x\right )=\frac{3x^{2}-2x}{x}\textup{ , what is the value of}f\left ( 4\right )\textup{?}$

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Answer

$\textup{Plug in 4 every time you see an }x\textup{ in the given equation and solve:}$

$f\left ( 4\right )=\frac{3\left ( 4\right )^{2}-2\left ( 4\right )}{4};;=\frac{48-8}{4}=10$

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Question

Use the function rule to find the for the following function:

$f(x) = 30+\frac{40}{x}$

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Answer

$f(x) = 30+\frac{40}{x}$

Given , , plug the value for x into the given equation and evaluate:

$f({\color{Red} 20}) = 30+\frac{40}{{\color{Red} 20}}$

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Question

Evaluate $\small f(g(3))$ if and .

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Answer

$\small f(g(3))$

This expression is the same as saying "take the answer of $\small g(3)$ and plug it into $\small f(x)$ ."

First, we need to find $\small g(3)$ . We do this by plugging $\small 3$ in for $\small x$ in $\small g(x)$ .

Now we take this answer and plug it into $\small f(x)$ .

We can find the value of $\small f(9)$ by replacing $\small x$ with $\small 9$ .

This is our final answer.

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Question

Let and . What is ?

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Answer

THe notation is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).

The original expression for f(x) is . We will take each x and substitute in the value of g(x), which is 2x-1.

We will now distribute the -2 to the 2x - 1.

We must FOIL the term, because .

Now we collect like terms. Combine the terms with just an x.

Combine constants.

The answer is .

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Question

Orange Taxi company charges passengers a $4.50 base fase, plus $0.10 per mile driven. Write a function to represent the cost of a cab ride, in terms of number of miles driven, .

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Answer

Total cost of the cab ride is going to equal the base fare ($4.50) plus an additional 10 cents per mile. This means the ride will always start off at $4.50. As the cab drives, the cost will increase by $0.10 each mile. This is represented as $0.10 times the number of miles. Therefore the total cost is:

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Question

A small office building is to be built with long walls feet long and short walls feet long each. The total length of the walls is to be feet.

Write an equation for in terms of .

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Answer

The pre-question text provides us with all of the information required to complete this problem.

We know that the total length of the walls is to be ft.

We also know that we have a total of walls and walls.

With this, we can set up an equation and solve for .

Our equation will be with sum of all the walls set equal to the total length of the wall...

Remeber, we want in terms of , which means our equation should look like

something

Subtract on both sides

Divide by on both sides

Simplify

Answer!!!

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Question

A cable company charges a flat $29.99 activation fee, and an additional $12.99 per month for service. How would a function of the cost be represented in terms of months of service, ?

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Answer

The flat rate of 29.99 does not change depending on months of service. It is $29.99 no matter how long services are in use. The monthy fee is directly related to the number of months the services are in use.

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Question

What is the slope of the function ?

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Answer

The function is written in slope-intercept form, which means:

where:

= slope

= x value

= y-intercept

Therefore, the slope is ${\color{Red} -5}$

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Question

If , and $g(x)=x^{2} -1$ , for which of the following value(s) will be an odd number?

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Answer

First, x needs to be plugged into g(x).

Then, the resulting solution needs to be substituted into f(x).

For example,

.

Since 45 is an odd number, 7 is an x value that gives this result. Because both equations subtract an odd number to get the final result, only an odd number will result in an odd result therefore, none of the other options will give an odd result.

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Question

Find for the following function:

$f(x)=\frac{4x+3}{x^2}$

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Answer

To evaluate , we just plug in a wherever we see an in the function, so our equation becomes

$f(3)=\frac{3\cdot 4+3}{3^2}$

which is equal to

$f(3)=\frac{5}{3}$

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Question

Find for the following function:

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Answer

To find , all we do is plug in wherever we see an in the function. We have to be sure we keep the parentheses. In this case, when we plug in , we get

$cos(5(g+h)^2)e^{g+h}$

Then, when we expand our binomial squared and distribute the , we get

$cos(5g^2+10gh+5h^2)e^{g+h}$

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Question

Evaluate the function for .

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Answer

To solve for the value of the function at , simply plug in the value in place of every . By doing this, you will be left with the equation:.

Another way to go about the problem is first simplifying the expression so that like terms are collected, so . Then to find , simply plug in the value in place of every . By doing this, you will be left with the equation:.

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Question

If , find .

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Answer

If then can be rewritten as . Therefore, . Subtracting from both sides of the eqauation gives .

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Question

Given and , find .

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Answer

Given f(x) and g(x), find f(g(5))

This type of problem can look intimidating depending on how it is set up. What it is asking is for us to plug g(x) into f(x) everywhere we see an x, and then to plug in 5 everywhere we still have an x. It gets a little cumbersome if approached all at once:

This looks a bit unwieldy, but this problem can be approached easily by looking at it in layers.

First, find g(5)

.

Next, plug that 15 into f(x).

So our answer is:

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