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Evaluating a Function

Author: Sophia
PDF Version

what's covered
This tutorial covers defining a function, through the definition and discussion of:

Table of Contents

1. Functions

A function is a relation between two variables in which an input variable corresponds to exactly one output variable. These variables can be referred to as inputs and outputs because the value for one variable is "put in" to a function, and then the function provides a specific set of operations to "output" a specific value for another variable.

EXAMPLE

Suppose you have the following rule, which would read as: "Take the input, multiply by 2, and add 3."



Evaluate the function for the following values.

Input Output Explanation
x equals 5 table attributes columnalign left end attributes row cell f open parentheses 5 close parentheses equals 5 times 2 plus 3 end cell row cell f open parentheses 5 close parentheses equals 10 plus 3 end cell row cell f open parentheses 5 close parentheses equals 13 end cell end table If you use the input 5 with this rule, the output would be 13 because 5 times 2 is 10 and 10 plus 3 is 13.
x equals 8 table attributes columnalign left end attributes row cell f open parentheses 8 close parentheses equals 8 times 2 plus 3 end cell row cell f open parentheses 8 close parentheses equals 16 plus 3 end cell row cell f open parentheses 8 close parentheses equals 19 end cell end table Similarly, if you use the input 8 with this rule, the output would be 19 because 8 times 2 is 16 and 16 plus 3 is 19.
x equals 5 table attributes columnalign left end attributes row cell f open parentheses short dash 10 close parentheses equals open parentheses short dash 10 close parentheses times 2 plus 3 end cell row cell f open parentheses short dash 10 close parentheses equals short dash 20 plus 3 end cell row cell f open parentheses short dash 10 close parentheses equals short dash 17 end cell end table Finally, if you use the input -10, the output would be -17 because -10 times 2 is -20 and -20 plus 3 is -17.

term to know
Function
A relation in which every element in the domain corresponds to exactly one element in the range

2. Interpreting Function Notation

The expression “f(x)” is commonly used for function notation. It is read "f of x" and does not mean f multiplied by x. Function notation is used to name a function where x is the independent variable or the input. The expression f(x) is used to represent the dependent variable or output of the function; therefore, it is the same as the variable y.

EXAMPLE

Another example of function notation would be g(t), where t is the independent variable or the input of the function g(t).

3. Evaluating Functions

To evaluate a function means to find the value of the output f(x) for a given input x. To evaluate a function, you replace each x in the function with the input value and use order of operations to simplify the expression to determine the output value.

EXAMPLE

Evaluate the function f open parentheses x close parentheses equals 4 x squared minus 6 for x equals 10.

First, replace each x with 10.

f left parenthesis 10 right parenthesis equals 4 left parenthesis 10 right parenthesis squared minus 6

Simplify starting with your exponent, then move on to multiplication and subtraction, which provides f(10) = 394.

table attributes columnalign left end attributes row cell f left parenthesis 10 right parenthesis equals 4 left parenthesis 100 right parenthesis minus 6 end cell row cell f left parenthesis 10 right parenthesis equals 400 minus 6 end cell row cell f left parenthesis 10 right parenthesis equals 394 end cell end table

Remember, you read this as f of 10, and it does not mean to multiply f and 10 together.

did you know
You can also represent your solution as an ordered pair — (10, 394) — which would be on the graph of this function.

try it
Consider the same function as above: f open parentheses x close parentheses equals 4 x squared minus 6.
Evaluate f(x) for x equals -4.
Start by replacing each x with -4.

table attributes columnalign left end attributes row cell f open parentheses x close parentheses equals 4 x squared minus 6 end cell row cell f open parentheses short dash 4 close parentheses equals 4 open parentheses short dash 4 close parentheses squared minus 6 end cell end table

Again, start with your exponent, then multiply and subtract.

table attributes columnalign left end attributes row cell f open parentheses short dash 4 close parentheses equals 4 open parentheses 16 close parentheses minus 6 end cell row cell f open parentheses short dash 4 close parentheses equals 64 minus 6 end cell row cell f open parentheses short dash 4 close parentheses equals 58 end cell end table

Again, you can write this as an ordered pair — (-4, 58) — which is a point that would be on the graph of the function f(x).

summary
Today you learned that a function is a relation between two variables—referred to as inputs and outputs—in which an input variable corresponds to exactly one output variable. You also learned that function notation is used to name a function where x is the independent variable or the input of the function. Lastly, you learned that f(x) is used to represent the dependent variable or output of the function, so it is the same as the variable y.

Source: This work is adapted from Sophia author Colleen Atakpu.

Terms to Know
Function

A relation in which every element in the domain corresponds to exactly one element in the range.

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