Unit 7: Exponential and General Functions

Topic 5

Composition and Inverses of Functions

Functions can be combined to form new ones, or reversed to find inverses. Composition means applying one function to the output of another:

(f \circ g)(x) = f(g(x)).

The order matters — generally

f(g(x)) \ne g(f(x))

An inverse function, written

f^{-1}(x)

, reverses the action of

f

. It “undoes”

f(x)

, so:

f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x.

To find an inverse algebraically:

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

A function has an inverse only if it is one-to-one (each input gives a unique output). Graphically, a function and its inverse are reflections across the line

y = x

Core Skills

Evaluate compositions like $f(g(x))$ and $g(f(x))$ .
Understand order in function composition.
Find inverses algebraically using variable swap and solve.
Recognize inverse pairs in tables or graphs.
Test if two functions are inverses using $f(g(x)) = x$ .

Example 1: Function Composition

f(x) = 2x + 3

and

g(x) = x - 1,

find:

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

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

\boxed{f(g(x)) = 2x + 1,\; g(f(x)) = 2x + 2}

Example 2: Composition with Quadratic Function

f(x) = x^2

and

g(x) = 3x + 2,

(f \circ g)(x) = f(3x + 2) = (3x + 2)^2 = 9x^2 + 12x + 4.

(g \circ f)(x) = g(x^2) = 3x^2 + 2.

\boxed{f(g(x)) = 9x^2 + 12x + 4,\; g(f(x)) = 3x^2 + 2}

Example 3: Finding an Inverse

Find the inverse of

f(x) = 3x - 5.

Step 1: Replace

f(x)

with

y

y = 3x - 5.

Step 2: Swap

x

and

y

x = 3y - 5.

Step 3: Solve for

y

3y = x + 5 \Rightarrow y = \frac{x + 5}{3}.

\boxed{f^{-1}(x) = \frac{x + 5}{3}}

Example 4: Verifying Inverses

Given

f(x) = 2x + 7

and

g(x) = \frac{x - 7}{2},

check if they are inverses. Compute

f(g(x))

f(g(x)) = 2\left(\frac{x - 7}{2}\right) + 7 = x - 7 + 7 = x.

and

g(f(x))

g(f(x)) = \frac{(2x + 7) - 7}{2} = \frac{2x}{2} = x.

\boxed{f \text{ and } g \text{ are inverses}}

Example 5: Nonlinear Inverse

Find the inverse of

f(x) = \dfrac{x + 4}{3}.

y = \frac{x + 4}{3} \quad \Rightarrow \quad x = \frac{y + 4}{3}.

Multiply both sides by 3:

3x = y + 4 \Rightarrow y = 3x - 4.

\boxed{f^{-1}(x) = 3x - 4}

Example 6: Graphical Idea (description only)

Graph

f(x) = 2x + 1

and its inverse

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

. They intersect at the point where

x = y

Key Takeaways

Composition means applying one function to another.
Order matters: $f(g(x)) \neq g(f(x))$ in general.
To find an inverse, swap $x$ and $y$ and solve for $y$ .
The graph of an inverse reflects across $y = x$ .
Verify inverses using composition: $f(g(x)) = g(f(x)) = x.$