How to Solve Composition of Functions? (+FREE Worksheet!)

Composition of functions is one of the most powerful tools in algebra: it lets you create a new function by feeding the output of one function directly into a second function. Written as (f ∘ g)(x) or f(g(x)), function composition appears throughout Algebra 1 and is the foundation for understanding inverse functions and transformations. With the right substitution method, every composition problem becomes straightforward.

What Is Composition of Functions?

When you compose two functions f and g, you apply one function and then immediately apply the other to the result. The notation f(g(x)) means “first apply g to x, then apply f to that result.” It is read “f of g of x.”

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• Inner function: g is applied first.
• Outer function: f is applied second to g’s output.

Important: Order matters. In general, f(g(x)) ≠ g(f(x)).

How to Find a Composition of Functions

Method — Substitute the inner function into the outer

Replace every x in the outer function’s formula with the entire inner function.

Example: \(\color{blue}{f(x) = 2x + 1}\), \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – 3}\). Find (f ∘ g)(x).

Step 1 — Write f, but replace every x with g(x):
\(\color{blue}{f(g(x)) = 2[g(x)] + 1 = 2(x^{2} – 3) + 1}\)
Step 2 — Distribute and simplify:
\(\color{blue}{= 2x^{2} – 6 + 1 = 2x^{2} – 5}\)

The reverse order: g(f(x))

Example (same f and g): Find (g ∘ f)(x).

Replace every x in g with f(x):
\(\color{blue}{g(f(x)) = [f(x)]^{2} – 3 = (2x + 1)^{2} – 3}\)
\(\color{blue}{= 4x^{2} + 4x + 1 – 3 = 4x^{2} + 4x – 2}\)

Evaluating at a specific value

You can substitute the inner function first, or work from the inside out.

Find (f ∘ g)(3): \(\color{blue}{g(3) = 9 – 3 = 6}\), then \(\color{blue}{f(6) = 2(6) + 1}\) = 13.

Step-by-Step Summary

1. Identify which function is the outer and which is the inner.
2. Write the outer function formula.
3. Replace every variable in the outer function with the inner function expression.
4. Expand and simplify (distribute, combine like terms).
5. If evaluating at a number, either substitute into the simplified formula or evaluate the inner function first and feed the result into the outer.

Watch: Introduction to Composition of Functions (Video Lesson)

Khan Academy introduces function composition with intuitive examples, explaining the inside-out structure of f(g(x)):

Composition of Functions – Worked Examples

Example 1: \(\color{blue}{f(x) = 2x + 1}\), \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – 3}\). Find (f ∘ g)(3).

Inside-out: \(\color{blue}{g(3) = (3)}\)² − \(\color{blue}{3 = 9 – 3 = 6}\)
Then: \(\color{blue}{f(6) = 2(6) + 1 = 13}\)
\(\color{blue}{(f &\#8728; g)(3) = 13}\)

Example 2: Same f and g. Find (g ∘ f)(2).

Inside-out: \(\color{blue}{f(2) = 2(2) + 1 = 5}\)
Then: \(\color{blue}{g(5) = (5)}\)² − \(\color{blue}{3 = 25 – 3 = 22}\)
\(\color{blue}{(g &\#8728; f)(2) = 22}\)

Example 3: Same f and g. Find (f ∘ g)(0).

\(\color{blue}{g(0) = 0 – 3}\) = −3; \(\color{blue}{f(-3) = 2(-3) + 1}\) = −5
\(\color{blue}{(f &\#8728; g)(0) = -5}\)

Example 4: Same f and g. Find (g ∘ f)(0).

\(\color{blue}{f(0) = 1}\); \(\color{blue}{g(1) = 1 – 3}\) = −2
\(\color{blue}{(g &\#8728; f)(0) = -2}\)

More Practice: Composite Functions Video

The Organic Chemistry Tutor covers composite functions with additional examples and tips for avoiding common mistakes:

Exercises for Composition of Functions

Let \(\color{blue}{f(x) = 2x + 1}\) and \(\color{blue}{g(x) = x}\)\(\color{blue}{^{2} – 3}\). Evaluate each composition.

1. (f ∘ g)(x) in simplified form
2. (g ∘ f)(x) in simplified form
3. (f ∘ g)(−2)
4. (g ∘ f)(−1)
5. (f ∘ f)(1)

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Answers

1. \(\color{blue}{(f &\#8728; g)(x) = 2(x^{2} – 3) + 1}\) = \(\color{blue}{2x^{2} – 5}\)
2. \(\color{blue}{(g &\#8728; f)(x) = (2x + 1)}\)\(\color{blue}{^{2} – 3}\) = 4x² + \(\color{blue}{4x – 2}\)
3. \(\color{blue}{g(-2) = 4 – 3 = 1}\); \(\color{blue}{f(1) = 3}\). Answer: 3
4. f(−1) = −1; \(\color{blue}{g(-1) = 1 – 3}\) = −2. Answer: −2
5. \(\color{blue}{f(1) = 3}\); \(\color{blue}{f(3) = 7}\). Answer: 7

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Free Composition of Functions Worksheet

Ready to practice on your own? Download our free Composition of Functions worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Composition of Functions before a quiz or test.

Download Combining Functions Worksheet

Frequently Asked Questions

Does the order of composition matter?

Yes. f(g(x)) and g(f(x)) usually give different results. The function closest to x is always applied first.

What is the domain of a composed function?

The domain of f(g(x)) is all x-values in the domain of g such that g(x) is also in the domain of f. You may need to check both conditions.

What does f(f(x)) mean?

It means composing f with itself: apply f once, then apply f again to that result. It is written (f ∘ f)(x).

Related Topics

• Function Notation
• Adding and Subtracting Functions
• Function Inverses
• Finding Domain and Range of a Function
• Multiplying and Dividing Functions