Composite Function Worksheet- Practice Problems with Solutions

What Are Composite Functions?

A composite function is what you get when you plug one function into another. If you have f(x) and g(x), the composite f(g(x)) means you first calculate g(x), then feed that result into f.

That's it. No magic, no complexity—just feeding outputs into inputs like a mathematical assembly line.

Notation to Know

You'll see these variations:

• (f ∘ g)(x) = f(g(x)) — apply g first, then f
• (g ∘ f)(x) = g(f(x)) — apply f first, then g

Order matters. f(g(x)) is not the same as g(f(x)) in most cases.

Evaluating Composite Functions: Step by Step

Here's how you actually solve these problems:

The Process

1. Start with the inner function — calculate it first
2. Take that result and use it as input for the outer function
3. Solve until you get your final answer

Quick Example

Given f(x) = 2x + 3 and g(x) = x², find f(g(2)).

Step 1: Evaluate g(2) = 2² = 4

Step 2: Evaluate f(4) = 2(4) + 3 = 8 + 3 = 11

Answer: f(g(2)) = 11

See how straightforward it is when you break it down? Most students overcomplicate this.

Practice Problems with Solutions

Work through each problem. Only check the solutions after you've attempted them.

Problem 1

Given f(x) = 3x - 5 and g(x) = x + 2, find f(g(x)).

Solution:

Replace x in f(x) with the entire g(x) expression:

f(g(x)) = 3(g(x)) - 5 = 3(x + 2) - 5 = 3x + 6 - 5 = 3x + 1

Problem 2

Given h(x) = x² - 4 and k(x) = √x, find k(h(3)).

Solution:

Step 1: h(3) = 3² - 4 = 9 - 4 = 5

Step 2: k(5) = √5

Answer: √5

Problem 3

If p(x) = 2x + 1, q(x) = 5x - 3, find (p ∘ q)(4).

Solution:

First find p(q(x)):

p(q(x)) = 2(5x - 3) + 1 = 10x - 6 + 1 = 10x - 5

Now plug in x = 4:

10(4) - 5 = 40 - 5 = 35

Problem 4

Given f(x) = 1/(x+1) and g(x) = x - 3, find f(g(x)) and state its domain.

Solution:

f(g(x)) = 1/((x-3) + 1) = 1/(x-2)

Domain restriction: denominator cannot be zero, so x ≠ 2

Problem 5

If a(x) = x + 4 and b(x) = x², compare a(b(x)) and b(a(x)).

Solution:

a(b(x)) = x² + 4

b(a(x)) = (x + 4)² = x² + 8x + 16

These are not equal. This proves composition isn't commutative.

Common Mistakes That Cost You Points

• Reversing the order — Always check which function is on the "outside"
• Skipping parentheses — f(g(x)) means substitute g(x) everywhere x appears in f
• Ignoring domain restrictions — composite functions inherit restrictions from both original functions
• Over-complicating simple problems — these are just substitution problems

Quick Reference: Composite Function Rules

How to Use These Worksheet Problems Effectively

Don't just read through the solutions. That's passive and useless for actually learning this.

Study Method That Works

1. Cover the solution
2. Attempt the problem with pencil and paper
3. Check your work
4. If wrong, find exactly where you went off track
5. Redo it correctly without looking

Repeat until you can solve each type without hints. Most students need 10-15 varied problems before this clicks.

When to Move On

You're ready to stop practicing when you can:

• Evaluate f(g(x)) given two functions
• Calculate numerical values like f(g(3))
• Find the domain of a composite function
• Explain why f(g(x)) ≠ g(f(x))

Finding More Practice Problems

Once you've mastered these basics, look for worksheets that include:

• Three-function compositions (f ∘ g ∘ h)
• Inverse composite functions
• Real-world word problems
• Graphical interpretation questions

Your textbook's problem sets are usually better than random online worksheets. Start there.