Nonlinear functions | Lesson (article) | Khan Academy (2024)

A function takes an input and produces an output. In function notation, $f(x)$‍, $f$‍ is the name of the function, $x$‍ is the input variable, and $f(x)$‍ is the output.

For example, given $f(x)=2x+1$‍, the expression $2x+1$‍ 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.

In this lesson, we'll learn to:

1. Evaluate functions algebraically
2. Determine inputs and outputs using tables
3. Evaluate

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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 $f(x)=x^3-9$‍, what is the value of $f(-2)$‍ ?

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 $f(x)=2x+1$‍.

Sometimes, a table of input-output pairs is provided without an algebraic function. Consider the table below.

The table contains four input-output pairs. We can interpret the information in the table as:

• $g(-2)=7$‍
• $g(-1)=0$‍
• $g(0)=4$‍
• $g(1)=-1$‍

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:

Based on the table above, what is the value of $f(3)$‍ ?

A composite function uses the output of one function as the input of another. For example, for $f(g(x))$‍:

• $x$‍ is the input to function $g$‍.
• $g(x)$‍ is the input to function $f$‍.

To evaluate composite functions at a specific input value:

1. Plug in the input value for the input variable wherever it appears in the .
2. Perform the operations specified by the inner function to calculate the output. This output becomes the input of the .
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 $f(x)=2x+1$‍ and $g(x)=x^2-2x+1$‍, what is the value of $f(g(-1))$‍ ?

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 $f(x)=x^2$‍ and $g(x)=x+1$‍:

From the table, we can tell that $g(1)=2$‍, and $f(2)=4$‍. Therefore, $f(g(1))=4$‍.

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:

The table above provides the values of functions $f$‍ and $g$‍ at several values of $x$‍. What is the value of $g(f(2))$‍ ?

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 $f(x)=x^2$‍ and $g(x)=x+1$‍, $f(g(x))$‍ replaces each instance of $x$‍ in $f$‍ with $g(x)$‍, which is equal to $x+1$‍: $f(g(x))=(x+1{)}^2$‍. Inputting an expression into a function, e.g., $f(x+1),$‍ works similarly.

A function can also be defined in terms of another function. For example, for $f(x)=x^2$‍ and $g(x)=f(x)+1$‍, we can replace the $f(x)$‍ in function $g$‍ with $x^2$‍: $g(x)=x^2+1$‍.

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.

If $f(x)=x-1$‍ and $g(x)=x^2+1$‍, what is $f(g(x))$‍ ?

If $g(x)=x^2+1$‍, what is $g(x-1)$‍ ?