Evaluating Nonlinear Functions | Grade 8 Math | LevelKite

A function is a rule that gives each permitted input exactly one output. The input is the number you put into the rule, and the output is the number the rule gives back. The domain is the set of all permitted inputs.

To evaluate a function, you find the output that goes with one input you choose from the domain.

You already know how to tell linear functions from nonlinear ones. A linear function has a rule you can write as $y = mx + b$ , where $x$ is the input, $y$ is the output, and $m$ and $b$ are fixed numbers ( $b$ can be zero or any other number). A nonlinear function is a function whose rule you cannot write that way. In this lesson, you will evaluate nonlinear functions.

Start with a rule you know. A square has a side length of $s$ centimeters (cm) and an area of $A$ square centimeters, where $A = s^2$ . The small raised $2$ is an exponent. It tells you to use $s$ as a factor twice, so $s^2 = s \cdot s$ .

Here the input is $s$ and the output is $A$ . A side length must be positive, so the domain is $s > 0$ .

To evaluate this function at $s = 3$ , replace $s$ with $3$ : $A = (3)^2 = 3 \cdot 3 = 9$ . A square with a side length of $3$ cm has an area of $9$ square centimeters. The table evaluates this input and three more in the same way.

Evaluating the area rule at four side lengths

Input: side length s (cm)	Substitute and calculate	Output: area A (square cm)
$1$	$A = (1)^2 = 1 \cdot 1$	$1$
$2$	$A = (2)^2 = 2 \cdot 2$	$4$
$3$	$A = (3)^2 = 3 \cdot 3$	$9$
$4$	$A = (4)^2 = 4 \cdot 4$	$16$

Look at the outputs in the table: $1$ , $4$ , $9$ , $16$ . Each time the input goes up by $1$ , the output goes up by $3$ , then $5$ , then $7$ . The output of a linear function would change by the same amount each time the input goes up by $1$ , so these unequal jumps show that this function is nonlinear.

That matters when you evaluate. For this function, you cannot find a new output by adding a fixed amount to the last one. You go back to the rule every time.

Key concept

Substitute, then follow the order of operations

When you have the rule for a function, evaluate it at an input in three steps:

Check that the input is in the domain.
Substitute: replace every copy of the input variable (the letter that stands for the input) with the value you chose. Put the value in parentheses so that a negative sign or an exponent stays attached to the whole value.
Calculate using the order of operations: inside parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

A function gives each permitted input exactly one output, so these steps lead to exactly one answer. Two different inputs may still share an output.

Worked example

A positive input

A function has the rule $y = 2x^2 + 1$ , where $x$ is the input and $y$ is the output. The input $x$ can be any real number, which means any number on the number line. Evaluate the function at $x = 3$ .

Check the domain: 3 is a real number, so it is a permitted input. The rule holds for every permitted input, so it still holds when you replace x with 3. Put the 3 in parentheses so it stays separate from the 2.
$y = 2(3)^2 + 1$
Work out the exponent first, because exponents come before multiplication. The exponent belongs only to the 3, and 3 squared means 3 times 3, which is 9.
$y = 2(9) + 1$
Multiply next, because multiplication comes before addition: 2 times 9 is 18.
$y = 18 + 1$
Add last: 18 plus 1 is 19.
$y = 19$

Answer

When $x = 3$ , the output of the function is $y = 19$ .

Worked example

A negative input

A function has the rule $y = x^2 - 4x$ , where $x$ is the input and $y$ is the output. The input $x$ can be any real number. Evaluate the function at $x = -3$ .

Negative 3 is a real number, so it is in the domain. The rule uses x twice, so replace both copies with negative 3. The parentheses keep the negative sign attached to the input.
$y = (-3)^2 - 4(-3)$
Work out the exponent first. Squaring negative 3 means multiplying negative 3 by itself, and a negative number times a negative number is positive.
$(-3)^2 = (-3)(-3) = 9$
Multiply next: 4 times negative 3. A positive number times a negative number is negative.
$4(-3) = -12$
Put both results back into the rule. Subtracting negative 12 is the same as adding 12, so add 9 and 12.
$y = 9 - (-12) = 9 + 12 = 21$

Answer

When $x = -3$ , the output of the function is $y = 21$ .

The rule from the last worked example, evaluated at five inputs

Input x	Substitute and calculate	Output y
$-3$	$y = (-3)^2 - 4(-3) = 9 - (-12)$	$21$
$0$	$y = (0)^2 - 4(0) = 0 - 0$	$0$
$2$	$y = (2)^2 - 4(2) = 4 - 8$	$-4$
$4$	$y = (4)^2 - 4(4) = 16 - 16$	$0$
$7$	$y = (7)^2 - 4(7) = 49 - 28$	$21$

Read down the output column. The inputs $-3$ and $7$ are different, yet both give the output $21$ . The inputs $0$ and $4$ both give $0$ . That is fine for a function: different inputs may share an output. What a function never does is give one input two different outputs. You can see that here, because each input in the table has just one output.

Worked example

A nonlinear rule with a restricted domain

A rectangle has an area of $24$ square centimeters and a width of $w$ centimeters. Its length $L$ , in centimeters, is the area divided by the width: $L = \frac{24}{w}$ . A width must be positive, so the domain is $w > 0$ . Find the length when the width is $5$ cm, and again when the width is $7$ cm.

Check the domain. Both 5 and 7 are greater than 0, so both are permitted inputs. Zero is not permitted: a rectangle cannot have a width of 0, and you cannot divide by 0.
Replace w with 5. The fraction bar means divide, and 24 divided by 5 is exactly 4.8, because 5 times 4.8 is 24.
$L = \dfrac{24}{5} = 4.8$
Replace w with 7. Dividing 24 by 7 gives a decimal that never ends, so the fraction itself is the exact output.
$L = \dfrac{24}{7}$
For a length you can measure, divide and round to the nearest tenth. The hundredths digit is 2, which is less than 5, so the tenths digit stays 4. A rounded value is an approximation, so write the approximately-equal sign in place of the equals sign.
$\dfrac{24}{7} = 3.428\ldots \approx 3.4$

Answer

When the width is $5$ cm, the length is exactly $4.8$ cm. When the width is $7$ cm, the length is exactly $\frac{24}{7}$ cm, which is about $3.4$ cm.

Takeaway: To evaluate a nonlinear function, choose an input from the domain, substitute it for the variable, and follow the order of operations, working out exponents before you multiply. You get exactly one output for that input, although a different input may give the same output. The outputs of a nonlinear function do not have to change by a fixed amount, so go back to the rule for each new input.