# Evaluating composite functions using tables

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## Function notation # Function notation on the SAT

## Studying function notation

On the SAT test function notation is part of passport to advanced mathematics subscore that includes 9 advanced topics (see the full topics list on the top menu).

Function notation topic is the last topic of passport to advances mathematics subscore. It is recommended to start learning passport to advances mathematics subscore with its first eight topics (the first topic is quadratic equations and quadratic functions).

Function notation topic is divided into sections from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.

Finish studying heart of algebra subscore topics before you study this topic or any other passport to advances mathematics subscore topic. (Heart of algebra subscore includes basic algebra topics which knowledge is required for understanding passport to advanced mathematics subscore topics).

## Function notation - summary

A function is a formula that represents relationship between input variable x (also called independent variable) and output variable f(x) (also called dependent variable). For example: f(x)=2x+5.

In function notation we evaluate the function, meaning finding the output of the function f(x) for a given input x. The input can be numeric (a number), an expression or another function.

An example for a function notation with a numeric input: the value of a function f(x)= 2x+5 when x=2 is f(2)=2x+5=2*2+5=9.
An example for a function notation with an  expression input: the value of a function f(x)= 2x+5 when f(x-1) is f(x-1)=2(x-1)+5=2x-2+5=2x+3.
An example for a function notation with a function g(x) as an input in another function f(x): the value of a function f(x)= 2g(x)+5 when g(x)=2x is f(g(x))=2(2x)+5=4x+5.

Representing values in a table instead of evaluating a function:
Instead of evaluating a function using its formula we can receive a table providing pairs of inputs and outputs with or without seeing the formula that stands behind the table.

For example: Consider the table below.

x              f(x)
2               9
t+1           2t+7

The formula that is behind the table is f(x)= 2x+5.

### Evaluating a function at a numeric input

Since the input is a number, we can calculate the output of a function as a number.

There are 2 steps for evaluating a function using its formula:
Step 1: Plugging the input value instead of the input variable.
Step 2: Calculating the output of the function.

Consider the following example:

What is the value of the function f(x)=3-x when x=3?

Step 1: Plugging the input value 3 instead of the input variable x:
Remember the negative exponents formula x-n=1/xn
f(x)=3-x

Step 2: Calculating the output of the function:
f(3)=3-3=1/33=1/(3*3*3)=1/27

Consider the following example:

The function f(x) is f(x)=(x-1)(x+2).

What is the value of f(2)-f(1)?

f(2)=(x-1)(x+2)=1*4=4

f(1)=0

f(2)-f(1)=4-0=4

### Evaluating a function with an expression input

In these questions the input is an expression instead of a single number, therefore in step 1 we need the plug the expression into the function the same way we plug the number.

Consider the following example:

If f(x)=(x+2)2 what is f(x-1)?

Step 1- Plugging the input expression x-1 instead of the input variable x:
f(x-1)=(x-1+2)2=(x+1)2

Step 2: Calculating the output of the function:
Remember square of sum formula: (a+b)2=a2+b2+2ab
(x+1)2=x2+1+2x=x2+2x+1

### Evaluating a function output using a table

In these questions we receive a table providing pairs of inputs and outputs without seeing the formula that stands behind the table.

There are 2 steps for evaluating a function output using a table:

Step 1: Finding the given input value in the input column of the table (the heading of the input column is x).

Step 2: Finding the output value that corresponds to the input in the output column of the table (the heading of the output column is f(x)).

Consider the following example:

The table below contains 3 input-output pairs for a function.

x    f(x)
-2    2
0    -2
1    -1

What is the value of f(-2)?

Step 1: Finding the input -2 in the input column of the table (the left column x)

The input -2 is written in the first row.

Step 2: Finding the output value that corresponds to the input -3 it the output column of the table (the right column f(x)).

Since we found that the given input is in the first row of the table, we need to look for the corresponding output in the first row of the table. The output that is written in the first row and a second column f(x) of the table is 2. Therefore f(-2)=2.

Note that you need to look for the input -2 in the left x column and not in the right f(x) column. If you confuse between the columns you will get a wrong answer that is 0.

### Composing 2 functions into a composite function

A composite function is a function that depends on another function as an input instead of a single variable x. For example: f(g(x)) is a composite function in which g(x) is substituted for x in f(x).

Note that f(g(x)) in different from g(f(x)). In order to evaluate f(g(x)) we need to substitute g(x) in f(x) while in order to evaluate g(f(x)) we need to substitute f(x) in g(x).

The steps for writing a composite function f(g(x)):

Step 1- evaluate the value of the outer function f(x) with the input x=g(x) by plugging the function of g(x) in f(x) instead of x.

Step 2- simplify the outer function: open brackets and combine like terms.

Consider the following example:

if f(x)=2x+3 and g(x)=4x+1

What is the value of f(g(x))?
What is the value of g(f(x))?

f(g(x))=2(4x+1)+3=8x+2+3=8x+5 g(f(x))=4(2x+3)+1=8x+12+1=8x+13

### Evaluating composite functions at a given input value

In these questions we are given the value of x, therefore we can calculate the numeric value of the composite function.

The steps for evaluating a composite function f(g(x)) at a given input value:

Step 1- calculate the value of the inner function g(x) with the given input x.

Step 2- calculate the value of the outer function f(x) with the input x=g(x) that was calculated in step 1.

Another way is to compose f(g(x)) first and then plug the given x into the composed function. This option can take longer therefore is less recommended.

Consider the following example:

if g(x)=2x+3 and f(x)=4x+1 what is the value of f(g(-1))?

We need to calculate the inner function g(x) first and then plug the answer into the outer function f(x).

Step 1– calculate the value of the inner function g(x):
g(x)=2x+3
g(-1)=2*-1+3=1

Step 2– calculate the value of the outer function f(x):
f(x)=4x+1
f(1)=4*1+1=5

Another way is to compose f(g(x)) first and then plug the given x into the composed function:
f(g(x))=4(2x+3)+1=8x+13
f(g(-1))=8*-1+13=-8+13=5

The answer is the same in both ways.

### Evaluating composite functions using tables

The table of a composite function has 3 columns instead of 2:
input column– x
output column of the function f(x) when we plug x into f(x)
output column of the function g(x) when we plug x into g(x)

We need to find the relevant values in the table: the given input is used as an input of the inner function while the output of the inner function from the table is used as an input for the outer function. Meaning that we must use the table twice with 2 different inputs and 2 different outputs.

The steps for evaluating composite function f(g(x)) using tables:
Step 1: Finding the given input value in the input column x.
Step 2: Finding the output value that corresponds to the given input x in the column of the inner function (the heading is g(x)).
Step 3: Finding the output value of the inner function g(x) that was calculated in step 2 in the input column x.
Step 4: Finding the output value that corresponds to the output value of the inner function g(x) in the column of the outer function (the heading is f(x)).

Consider the following example:

The table below provides the values of functions f(x) and g(x) at several values of x.

Calculate is the value of f(g(1)), f(g(0)), g(f(-1)) and g(f(-2)).

x    g(x)   f(x)
-2      4       0
-1      1       2
0      0       4
1      1       6
2      4       8

Calculating the value of f(g(1)):

The steps are marked in the headings and in the table below:

Step 1: Finding the given input 1 in the input column x
The row is the fourth row.

Step 2: Finding the output value that corresponds to the input 1 in the column of the inner function g(x).
The column is the second column and the output is g(1) is 1.

Step 3: Finding the output 1  in the input column x
The row is the fourth row.

Step 4: Finding the output value in the column of the outer function f(x)
The column of the outer function f(x) is the third column. The input of the inner function g(1)=1 is located in the fourth row therefore the output of the outer function f(x) is 6. Calculating the value of f(g(0)):

The steps are marked in the headings and in the table below:

Step 1: Finding the given input 0 in the input column x
The row is the third row.

Step 2: Finding the output value that corresponds to the input 0 in the column of the inner function g(x).
The column is the second column and the output is g(0) is 0.

Step 3: Finding the output 0  in the input column x
The row is the third row.

Step 4: Finding the output value in the column of the outer function f(x)
The column of the outer function f(x) is the third column. The input of the inner function g(0)=0 is located in the third row therefore the output of the outer function f(x) is 4. Calculating g(f(-1)):

The steps are marked in the headings and in the table below:

Step 1: Finding the given input -1 in the input column x

The row is the second row.

Step 2: Finding the output value that corresponds to the input -1 in the column of the inner function f(x).

The column is the third column and the output is f(-1) is 2.

Step 3: Finding the output 2 in the input column x

The row is the fifth row.

Step 4: Finding the output value in the column of the outer function g(x)

The column of the outer function g(x) is the second column. The input of the inner function f(-1)=2 is located in the fifth row therefore the output of the outer function f(x) is 4. Calculating g(f(-2)):

The steps are marked in the headings and in the table below:

Step 1: Finding the given input -2 in the input column x

The row is the first row.

Step 2: Finding the output value that corresponds to the input -1 in the column of the inner function f(x).

The column is the third column and the output is f(-2) is 0.

Step 3: Finding the output 0 in the input column x

The row is the third row.

Step 4: Finding the output value in the column of the outer function g(x)

The column of the outer function g(x) is the second column. The input of the inner function f(-2)=0 is located in the third row therefore the output of the outer function f(x) is 0. 