Composite and Inverse Functions

Combining and reversing functions

Composite Functions

What is a Composite Function?

A composite function is created when one function is applied to the result of another function. It's like putting the output of one machine into another machine.

Notation: \(fg(x)\) or \(f(g(x))\) means "apply g first, then apply f to the result"

Composite Function Notation

\[fg(x) = f(g(x))\]

Read as "f of g of x" — work from the inside out!

Example 1: Finding fg(x)

Given \(f(x) = 2x + 3\) and \(g(x) = x^2\), find \(fg(x)\)

1 Start with g(x):

\[g(x) = x^2\]

2 Substitute g(x) into f:

\[fg(x) = f(g(x)) = f(x^2)\]

3 Replace x in f(x) with x²:

\[f(x^2) = 2(x^2) + 3 = 2x^2 + 3\]

Answer: \(fg(x) = 2x^2 + 3\)

Example 2: Finding gf(x) — Order Matters!

Using the same functions \(f(x) = 2x + 3\) and \(g(x) = x^2\), find \(gf(x)\)

1 Start with f(x):

\[f(x) = 2x + 3\]

2 Substitute f(x) into g:

\[gf(x) = g(f(x)) = g(2x + 3)\]

3 Replace x in g(x) with (2x + 3):

\[g(2x + 3) = (2x + 3)^2 = 4x^2 + 12x + 9\]

Answer: \(gf(x) = 4x^2 + 12x + 9\)

Notice: \(fg(x) \neq gf(x)\) — the order matters!

Example 3: Evaluating a Composite

Given \(f(x) = x + 4\) and \(g(x) = 3x\), find \(fg(2)\)

Method 1: Step by step

First find \(g(2) = 3(2) = 6\) Then find \(f(6) = 6 + 4 = 10\) So \(fg(2) = 10\)

Method 2: Find fg(x) first

\(fg(x) = f(3x) = 3x + 4\) \(fg(2) = 3(2) + 4 = 10\)

Interactive Composite Function Machine

Function Composition Visualizer

f(x) =

g(x) =

x =

Input

→

g(x)

x - 2

→

g(x) output

→

f(x)

2x + 1

→

fg(x) output

\(fg(x) = f(g(x)) = f(x-2) = 2(x-2) + 1 = 2x - 3\)

fg(3) = 3

Inverse Functions

What is an Inverse Function?

The inverse function \(f^{-1}(x)\) "undoes" what the original function \(f(x)\) does. If \(f\) takes you from A to B, then \(f^{-1}\) takes you from B back to A.

Key Property: \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\)

f(x)

→

f⁻¹(x)

←

Finding the Inverse Function

Replace \(f(x)\) with \(y\)
Swap \(x\) and \(y\)
Solve for \(y\)
Replace \(y\) with \(f^{-1}(x)\)

Example 4: Finding an Inverse

Find the inverse of \(f(x) = 3x + 5\)

1 Replace f(x) with y:

\[y = 3x + 5\]

2 Swap x and y:

\[x = 3y + 5\]

3 Solve for y:

\[x - 5 = 3y\]

\[y = \frac{x - 5}{3}\]

4 Write the inverse:

\[f^{-1}(x) = \frac{x - 5}{3}\]

Verify: \(f(f^{-1}(x)) = f\left(\frac{x-5}{3}\right) = 3 \cdot \frac{x-5}{3} + 5 = x - 5 + 5 = x\) ✓

Example 5: More Complex Inverse

Find the inverse of \(f(x) = \frac{2x + 1}{3}\)

1 Write as y: \(y = \frac{2x + 1}{3}\)

2 Swap: \(x = \frac{2y + 1}{3}\)

3 Solve:

\(3x = 2y + 1\)

\(3x - 1 = 2y\)

\(y = \frac{3x - 1}{2}\)

Answer: \(f^{-1}(x) = \frac{3x - 1}{2}\)

Important Notes

\(f^{-1}\) does NOT mean \(\frac{1}{f}\) — it's inverse notation, not a power!
Not all functions have inverses (only one-to-one functions do)
The graph of \(f^{-1}\) is the reflection of \(f\) in the line \(y = x\)

Practice Problems

Question 1: Given \(f(x) = x + 2\) and \(g(x) = 3x\), find \(fg(4)\)

Show Solution

\(g(4) = 3(4) = 12\)

\(f(12) = 12 + 2 = 14\)

\(fg(4) = 14\)

Question 2: Given \(f(x) = 2x - 1\) and \(g(x) = x + 3\), find \(gf(x)\)

Show Solution

\(gf(x) = g(f(x)) = g(2x - 1)\)

\(= (2x - 1) + 3 = 2x + 2\)

Question 3: Find the inverse of \(f(x) = 4x - 7\)

Show Solution

\(y = 4x - 7\)

\(x = 4y - 7\)

\(x + 7 = 4y\)

\(y = \frac{x + 7}{4}\)

\(f^{-1}(x) = \frac{x + 7}{4}\)

Question 4: If \(f(x) = 5x + 2\), find \(f^{-1}(17)\)

Show Solution

First find \(f^{-1}(x)\):

\(y = 5x + 2 \Rightarrow x = 5y + 2 \Rightarrow y = \frac{x-2}{5}\)

\(f^{-1}(x) = \frac{x-2}{5}\)

\(f^{-1}(17) = \frac{17-2}{5} = \frac{15}{5} = 3\)

Question 5: Given \(f(x) = x^2\) and \(g(x) = x + 1\), find \(fg(x)\) and \(gf(x)\)

Show Solution

\(fg(x) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1\)

\(gf(x) = g(x^2) = x^2 + 1\)

CSEC Quick Reference

Composite: \(fg(x)\) = apply g first, then f

Inverse: swap x and y, solve for y

Check: \(f(f^{-1}(x)) = x\)

Remember: \(fg(x) \neq gf(x)\) in general!