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Nonlinear functions | Lesson

A guide to nonlinear functions on the digital SAT

What are nonlinear functions problems?

What is a function?

Khan Academy video wrapper
What is a function?See video transcript
A function takes an input and produces an output. In function notation, f, left parenthesis, x, right parenthesis, f is the name of the function, x is the input variable, and f, left parenthesis, x, right parenthesis is the output.
For example, given f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 1, the expression 2, x, plus, 1 works as instructions on what to do with the input x. In this case, the input x is multiplied by 2, then 1 is added to the product.
The input of a function can be a
numeric value
, an
expression
, or even
another function
. Functions can also be
based on other functions
.
In this lesson, we'll learn to:
  1. Evaluate functions algebraically
  2. Determine inputs and outputs using tables
  3. Evaluate
    composite functions
  4. Compose two functions
You can learn anything. Let's do this!

How do I evaluate functions?

Evaluate a function given its formula

Khan Academy video wrapper
Worked example: Evaluating functions from equationSee video transcript

Evaluating functions algebraically and using tables

When we encounter an algebraic function, we can find the value of the function at specific inputs. For example, for f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 1, we can calculate f, left parenthesis, 2, right parenthesis, the output of the function f when its input, x, is equal to 2.

What are the steps?

To evaluate a function at a specific input value:
  1. Plug in the input value for the input variable wherever it appears.
  2. Perform the operations specified by the function to calculate the ouput.

Example: If f, left parenthesis, x, right parenthesis, equals, x, cubed, minus, 9, what is the value of f, left parenthesis, minus, 2, right parenthesis ?

With algebraic functions, we can evaluate the function using multiple inputs to create multiple input-output pairs. These input-output pairs can be put in a table, as shown below for f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 1.
xf, left parenthesis, x, right parenthesis
01
13
25
37
Sometimes, a table of input-output pairs is provided without an algebraic function. Consider the table below.
xg, left parenthesis, x, right parenthesis
minus, 27
minus, 10
04
1minus, 1
The table contains four input-output pairs. We can interpret the information in the table as:
  • g, left parenthesis, minus, 2, right parenthesis, equals, 7
  • g, left parenthesis, minus, 1, right parenthesis, equals, 0
  • g, left parenthesis, 0, right parenthesis, equals, 4
  • g, left parenthesis, 1, right parenthesis, equals, minus, 1
To evaluate a function using a table:
  1. Find the input value you're looking for in the input column (typically the left column with a header of the input variable such as x).
  2. Find the corresponding output value in the output column.

Example:
xf, left parenthesis, x, right parenthesis
03
12
35
43
Based on the table above, what is the value of f, left parenthesis, 3, right parenthesis ?

Try it!

try: complete a table based on a function
f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 3
Complete the table below.
xf, left parenthesis, x, right parenthesis
0
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
1minus, 2
2
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
4
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


How do I evaluate composite functions?

Intro to function composition

Khan Academy video wrapper
Intro to composing functionsSee video transcript

Evaluating composite functions algebraically and using tables

A composite function uses the output of one function as the input of another. For example, for f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis:
  • x is the input to function g.
  • g, left parenthesis, x, right parenthesis is the input to function f.
As such, composite functions should be worked from the inside out. Order matters when evaluating composite functions: g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis is not the same as f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis! For example, for f, left parenthesis, x, right parenthesis, equals, x, squared and g, left parenthesis, x, right parenthesis, equals, x, plus, 1, f, left parenthesis, g, left parenthesis, 1, right parenthesis, right parenthesis, equals, 4, but g, left parenthesis, f, left parenthesis, 1, right parenthesis, right parenthesis, equals, 2.
To evaluate composite functions at a specific input value:
  1. Plug in the input value for the input variable wherever it appears in the
    inner function
    .
  2. Perform the operations specified by the inner function to calculate the output. This output becomes the input of the
    outer function
    .
  3. Plug in the result of Step 2 for the input variable wherever it appears in the outer function.
  4. Perform the operations specified by the outer function to calculate the final output.

Example: If f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 1 and g, left parenthesis, x, right parenthesis, equals, x, squared, minus, 2, x, plus, 1, what is the value of f, left parenthesis, g, left parenthesis, minus, 1, right parenthesis, right parenthesis ?

Composite functions can also be evaluated using a table. The table can have an additional column for a total of three: one column for input and two columns for the outputs of two functions. Consider the table for f, left parenthesis, x, right parenthesis, equals, x, squared and g, left parenthesis, x, right parenthesis, equals, x, plus, 1:
xf, left parenthesis, x, right parenthesisg, left parenthesis, x, right parenthesis
start color #7854ab, 1, end color #7854ab1start color #ca337c, 2, end color #ca337c
start color #ca337c, 2, end color #ca337cstart color #208170, 4, end color #2081703
394
4165
From the table, we can tell that g, left parenthesis, start color #7854ab, 1, end color #7854ab, right parenthesis, equals, start color #ca337c, 2, end color #ca337c, and f, left parenthesis, start color #ca337c, 2, end color #ca337c, right parenthesis, equals, start color #208170, 4, end color #208170. Therefore, f, left parenthesis, g, left parenthesis, 1, right parenthesis, right parenthesis, equals, 4.
To evaluate composite functions at a specific input value given a table:
  1. Find the output value for the inner function corresponding to the specific input value. This is also the input value of the outer function.
  2. Find the output value for the outer function corresponding to the input of the result of Step 1.

Example:
xf, left parenthesis, x, right parenthesisg, left parenthesis, x, right parenthesis
130
251
374
499
51116
The table above provides the values of functions f and g at several values of x. What is the value of g, left parenthesis, f, left parenthesis, 2, right parenthesis, right parenthesis ?

Try it!

try: evaluate composite functions using a table
xp, left parenthesis, x, right parenthesis
01
12
23
35
The table above shows the value of function p at several values of x. Since p, left parenthesis, 2, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
and p, left parenthesis, 3, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
, the value of p, left parenthesis, p, left parenthesis, 2, right parenthesis, right parenthesis is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.


How do I compose functions?

Finding composite functions

Khan Academy video wrapper
Finding composite functionsSee video transcript

Inputting expressions instead of values into functions

In addition to inputting a specific value, we can also input one function into another function, which creates a composite function defined by a single expression.
For example, for f, left parenthesis, x, right parenthesis, equals, x, squared and g, left parenthesis, x, right parenthesis, equals, x, plus, 1, f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis replaces each instance of x in f with g, left parenthesis, x, right parenthesis, which is equal to x, plus, 1: f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, squared. Inputting an expression into a function, e.g., f, left parenthesis, x, plus, 1, right parenthesis, comma works similarly.
A function can also be defined in terms of another function. For example, for f, left parenthesis, x, right parenthesis, equals, x, squared and g, left parenthesis, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis, plus, 1, we can replace the f, left parenthesis, x, right parenthesis in function g with x, squared: g, left parenthesis, x, right parenthesis, equals, x, squared, plus, 1.
If you find yourself struggling to rewrite complex functions, you might want to brush up on the Operations with polynomials and Operations with rational expressions skills, which have their own lessons.
To compose two functions:
  1. Plug in the expression that defines the inner function wherever the input variable appears in the outer function.
  2. Perform the operations specified by the outer function. Combine like terms as needed.

Let's look at some examples!

If f, left parenthesis, x, right parenthesis, equals, x, minus, 1 and g, left parenthesis, x, right parenthesis, equals, x, squared, plus, 1, what is f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis ?

If g, left parenthesis, x, right parenthesis, equals, x, squared, plus, 1, what is g, left parenthesis, x, minus, 1, right parenthesis ?

Try it!

try: identify equivalent function composition
f(x)=x2+3x+7g(x)=x3\begin{aligned} f(x) &=x^2+3x+7 \\\\ g(x) &=x^3 \end{aligned}
The functions f and g are shown above. Which of the following expressions is equivalent to g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis ?
Choose 1 answer:
Choose 1 answer:


try: Draw conclusions about the features of two related functions
f(x)=x2+10g(x)=f(x)7\begin{aligned} f(x)&=-x^2+10 \\\\ g(x)&=f(x)-7 \end{aligned}
The functions f and g are shown above. For the same value of x, the value of g, left parenthesis, x, right parenthesis is always
the value of f, left parenthesis, x, right parenthesis.
Therefore, if the maximum value of the function f is 10, then the maximum value of function g is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.


Your turn!

TRY: evaluate a function at integer values
p, left parenthesis, x, right parenthesis, equals, start fraction, 2, divided by, x, end fraction
The function p is defined above. What is the value of p, left parenthesis, 2, right parenthesis, plus, p, left parenthesis, 5, right parenthesis ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


TRY: evaluate a composite function
If f, left parenthesis, x, right parenthesis, equals, 3, x, plus, 5 and g, left parenthesis, x, right parenthesis, equals, x, squared, minus, 4, what is the value of g, left parenthesis, f, left parenthesis, 1, right parenthesis, right parenthesis ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


TRY: evaluate a composite function using table of values
xh, left parenthesis, x, right parenthesis
minus, 37
minus, 24
10
3minus, 3
43
The table above shows the value of the function h at certain values of x. What is the value of h, left parenthesis, h, left parenthesis, 3, right parenthesis, right parenthesis ?
Choose 1 answer:
Choose 1 answer:


TRY: compose two functions
If f, left parenthesis, x, right parenthesis, equals, x, squared, minus, 2, x, minus, 5, which of the following is equivalent to f, left parenthesis, x, plus, 3, right parenthesis ?
Choose 1 answer:
Choose 1 answer:


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