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\( f(x) \)

Evaluating
Functions

Learn how to plug in a value for \( x \), compute the output, and interpret what \( f(x) \) means in real-world contexts.

Pre-Algebra

What Is a Function?

A function is a rule that takes an input and produces exactly one output. Think of it like a machine: you feed something in, and something specific comes out.

Input Function
\( f(x) = \text{rule} \)
 Output

Understanding \( f(x) \) Notation

The notation \( f(x) \) is read "f of x." It does not mean \( f \) times \( x \). It means: "the output of function \( f \) when the input is \( x \)."

\( f \)
The name of the function (could also be \( g \), \( h \), etc.)
\( x \)
The input variable: the value you plug in
\( f(x) \)
The output: the result after applying the rule

Why Evaluating Functions Matters

Functions model real-world situations. Evaluating them answers practical questions like:

1
Temperature: If \( T(h) \) gives the temperature at hour \( h \), what is \( T(6) \)?
2
Cost: If \( C(n) \) gives the cost of \( n \) items, what is \( C(12) \)?
3
Distance: If \( d(t) \) gives the distance traveled after \( t \) seconds, what is \( d(5) \)?

The Substitution Rule

To evaluate \( f(a) \), replace every \( x \) in the function rule with \( a \), then simplify.

Start with the rule
\( f(x) = 3x + 5 \)
 Replace x with 4
\( f(4) = 3(4) + 5 \)
 Simplify
\( f(4) = 17 \)
 Key Idea Every x becomes the given input

Functions as Tables

Sometimes a function is given as a table of values instead of a formula. To evaluate, just look up the input and read the output.

\( x \) 0 1 2 3 4
\( f(x) \) 7 10 13 16 19

Functions vs. Expressions

Students sometimes confuse evaluating a function with simplifying an expression. Here is the key difference:

Function Evaluation Expression Simplification
What you start with \( f(x) = 2x + 3 \), find \( f(5) \) Simplify \( 2x + 3x \)
What you do Substitute 5 for every \( x \) Combine like terms
Result A number: 13 An expression: \( 5x \)

Step-by-Step Method

1
Identify the input: What value are you plugging in? (e.g., in \( f(3) \), the input is 3)
2
Write the rule: Copy the function formula exactly as given.
3
Substitute: Replace every \( x \) with the input value. Use parentheses around it.
4
Simplify: Follow order of operations (PEMDAS) to get a single number.
5
Write the answer: State \( f(\text{input}) = \text{output} \).

Example 1: Linear Function

Evaluate \( f(x) = 4x - 7 \) at \( x = 3 \).

\( f(3) = 4(3) - 7 \)
\( f(3) = 12 - 7 \)
\( f(3) = 5 \)

Example 2: Negative Input

Evaluate \( g(x) = x^2 + 2x \) at \( x = -3 \).

\( g(-3) = (-3)^2 + 2(-3) \)
\( g(-3) = 9 + (-6) \)
\( g(-3) = 3 \)

Example 3: Real-World Context

A taxi charges a flat fee plus a per-mile rate. The total cost is modeled by \( C(m) = 2.5m + 3 \), where \( m \) is miles traveled. Find the cost for a 10-mile ride.

1
Identify the input: \( m = \) 10
2
Substitute: \( C(10) = 2.5(10) + 3 \)
3
Simplify: \( C(10) = 25 + 3 = \) 28

Evaluating from a Graph

You can also evaluate a function by reading its graph. Go to the input on the x-axis, move up (or down) to the curve, then read the y-value.

0 2 4 6 0 1 2 3 4 x
\( f(x) \)
\( f(4) = 3 \)

Three Ways to Evaluate

Functions can be represented in different forms. You evaluate them differently depending on the form:

Formula
Substitute the input into the equation and simplify
Table
Find the input in the top row, read the output below it
Graph
Locate the input on the x-axis, read the y-value on the curve

Inputs Can Be Expressions Too

Sometimes the input is not just a number: it can be an expression like \( a + 1 \) or \( 2t \). The same substitution rule applies.

1
Given \( f(x) = x^2 - 1 \), find \( f(a + 1) \).
2
Replace every \( x \) with \( (a + 1) \): \( f(a+1) = (a+1)^2 - 1 \)
3
Expand: \( (a+1)^2 - 1 = a^2 + 2a + 1 - 1 = a^2 + 2a \)

Common Mistakes to Avoid

✕ Treating \( f(x) \) as multiplication
\( f(3) \) does not mean \( f \times 3 \). It means "evaluate the function at 3."
✕ Forgetting parentheses
Writing \( -3^2 = -9 \) instead of \( (-3)^2 = 9 \). Always wrap negative inputs.
✕ Replacing only one \( x \)
In \( f(x) = x^2 + x \), you must replace both \( x \)'s when evaluating.

Summary Cheat Sheet

Concept What to Do Example
Evaluate at a number Replace \( x \) with the number \( f(2) = 2(2)+1 = 5 \)
Evaluate from a table Look up the input, read the output If row says 3 → 10, then \( f(3) = 10 \)
Evaluate from a graph Go to \( x \), read the \( y \)-value Point at \( (4, 3) \) means \( f(4) = 3 \)
Negative input Use parentheses: \( (-a) \) \( f(-2) = (-2)^2 = 4 \)
Expression input Replace \( x \) with the whole expression \( f(a+1) = (a+1)^2 \)

Question 1 of 5

If \( f(x) = 3x - 4 \), what is \( f(5) \)?

Question 2 of 5

If \( g(x) = x^2 + 3 \), what is \( g(-2) \)?

Question 3 of 5

The table below shows values for \( h(x) \). What is \( h(4) \)?

\( x \) 1 2 3 4 5
\( h(x) \) 5 9 13 17 21

Question 4 of 5

A movie rental service charges \( C(n) = 1.50n + 5 \) dollars, where \( n \) is the number of movies. What does \( C(6) = 14 \) tell us?

Question 5 of 5

If \( f(x) = 2x^2 - x + 1 \), what is \( f(0) \)?