Algebra · Step-by-step guide

Function Notation

Read and evaluate function notation, find inputs and outputs, and work with composite expressions.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

f(x)\displaystyle f(x) names the output of function f for input x; it does not mean f times x. To evaluate f(a)\displaystyle f(a), replace every x in the rule with a, using parentheses.

01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Input goes inside

The expression in parentheses tells you what replaces the independent variable.

2

Output is the result

An equation such as f(3) = 10 says the input 3 produces output 10.

3

Parentheses protect substitutions

They are essential when the input is negative or contains multiple terms.

02

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Number input

If f(x)=2x2−3\displaystyle f(x)=2x^2-3, find f(−2)\displaystyle f(-2).

  1. Replace x with −2 in parentheses.
  2. f(−2)=2(−2)2−3\displaystyle f(-2)=2(-2)^2-3.
  3. Evaluate the exponent before multiplying.
Answerf(−2)=5\displaystyle f(-2)=5
Example 2Expression input

If g(x)=3x+1\displaystyle g(x)=3x+1, find g(t−2)\displaystyle g(t-2).

  1. Replace x with t−2\displaystyle t-2.
  2. Distribute 3: 3(t−2)+1\displaystyle 3(t-2)+1.
  3. Combine constants.
Answerg(t−2)=3t−5\displaystyle g(t-2)=3t-5
Example 3Find the input

If h(x)=5x−4\displaystyle h(x)=5x-4 and h(x)=21\displaystyle h(x)=21, find x.

  1. Set the rule equal to 21.
  2. Solve 5x−4=21\displaystyle 5x-4=21.
  3. Then 5x=25\displaystyle 5x=25.
Answerx=5\displaystyle x=5
03

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Reading f(x) as multiplication.

  • Substituting into only one occurrence of x.

  • Squaring a negative input without parentheses.

04

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Function Notation

Practice evaluating functions with numbers and expressions.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

If f(x)=3x−4\displaystyle f(x)=3x-4, find f(5)\displaystyle f(5).

Show hint

Replace x with 5.

Show worked solution
  1. f(5)=3(5)−4\displaystyle f(5)=3(5)-4.
  2. 15−4=11\displaystyle 15-4=11.

Answer: 11\displaystyle 11

2

If g(t)=t2+2t\displaystyle g(t)=t^2+2t, find g(−3)\displaystyle g(-3).

Show hint

Use parentheses around −3.

Show worked solution
  1. g(−3)=(−3)2+2(−3)\displaystyle g(-3)=(-3)^2+2(-3).
  2. 9−6=3\displaystyle 9-6=3.

Answer: 3\displaystyle 3

3

If h(x)=2x+1\displaystyle h(x)=2x+1, solve h(x)=13\displaystyle h(x)=13.

Show hint

Set the formula equal to 13.

Show worked solution
  1. 2x+1=13\displaystyle 2x+1=13.
  2. 2x=12\displaystyle 2x=12.
  3. x=6\displaystyle x=6.

Answer: x=6\displaystyle x=6

4

Given f(x)=x2−1\displaystyle f(x)=x^2-1, simplify f(a+1)\displaystyle f(a+1).

Show hint

Substitute the entire expression.

Show worked solution
  1. f(a+1)=(a+1)2−1\displaystyle f(a+1)=(a+1)^2-1.
  2. Expand: a2+2a+1−1\displaystyle a^2+2a+1-1.

Answer: a2+2a\displaystyle a^2+2a

5

If f(x)=2x\displaystyle f(x)=2x and g(x)=x+3\displaystyle g(x)=x+3, find f(g(4))\displaystyle f(g(4)).

Show hint

Evaluate inside first.

Show worked solution
  1. g(4)=4+3=7\displaystyle g(4)=4+3=7.
  2. f(7)=2(7)=14\displaystyle f(7)=2(7)=14.

Answer: 14\displaystyle 14

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

f(x)\displaystyle f(x) names the output of function f for input x; it does not mean f times x. To evaluate f(a)\displaystyle f(a), replace every x in the rule with a, using parentheses.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

Keep learning

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