# Algebra II : Function Notation

## Example Questions

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### Example Question #1 : Understanding Functional Notations

Let  and . What is ?

Explanation:

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 .

### Example Question #1 : Function Notation

Solve the function for . When

What does  equal when,

0

25

-5

Explanation:

Plug 16 in for .

Add 9 to both sides.

Take the square root of both sides. =

### Example Question #1 : Function Notation

Evaluate  if and .

Undefined

Explanation:

This expression is the same as saying "take the answer of and plug it into ."

First, we need to find . We do this by plugging in for in .

Now we take this answer and plug it into .

We can find the value of by replacing with .

This is our final answer.

### Example Question #2 : Function Notation

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, .

Explanation:

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:

### Example Question #3 : Function Notation

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 .

Explanation:

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

### Example Question #4 : Function Notation

What is the slope of the function ?

Explanation:

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

where:

= slope

= x value

= y-intercept

Therefore, the slope is

### Example Question #1 : Function Notation

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, ?

This cannot be written as a function

Explanation:

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.

### Example Question #6 : Function Notation

Find  for the following function:

Explanation:

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

which is equal to

### Example Question #7 : Function Notation

Find  for the following function:

Explanation:

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

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

### Example Question #8 : Function Notation

If , and  , for which of the following  value(s) will  be an odd number?