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Mathematics

Composite & Inverse Functions

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Amari Cross & Matthew Williams
||5 min read
CompositeFunctionsInverse

Finding and evaluating composite functions, and how to find and verify an inverse.

Composite and inverse functions are about processes. A composite function performs one rule and then another; an inverse function reverses a rule and takes you back to the original input.

In CSEC, this topic sits in Relations, Functions and Graphs, where questions may ask you to evaluate, form, or rearrange functions. Pay close attention to order: f(g(x))f(g(x)) usually gives a different result from g(f(x))g(f(x)). For inverse functions, explain how each algebraic step undoes the original operation.

What Is a Composite Function?

A composite function is when you apply one function, then apply another function to the result.

Notation: (f∘g)(x)(f \circ g)(x) or f(g(x))f(g(x))

Read as: "ff composed with gg of xx"

Process:

  1. First, find g(x)g(x) (apply inner function)
  2. Then, apply ff to that result: f(g(x))f(g(x))

Important: f(g(x))≠g(f(x))f(g(x)) \neq g(f(x)) (order matters!)

Finding Composite Functions

Example

Given f(x)=2x+1f(x) = 2x + 1 and g(x)=x2g(x) = x^2, find f(g(x))f(g(x)):

Step 1: Identify what we're doing f(g(x))=apply g first, then apply ff(g(x)) = \text{apply } g \text{ first, then apply } f

Step 2: Find g(x)=x2g(x) = x^2

Step 3: Apply ff to the result: wherever we see xx in the definition of ff, substitute g(x)=x2g(x) = x^2 f(g(x))=f(x2)=2(x2)+1=2x2+1f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1

Answer: f(g(x))=2x2+1f(g(x)) = 2x^2 + 1

This is a new function that takes xx, squares it, multiplies by 2, and adds 1.

Example

Given f(x)=2x+1f(x) = 2x + 1 and g(x)=x2g(x) = x^2, find g(f(x))g(f(x)):

Step 1: Apply ff first to get f(x)=2x+1f(x) = 2x + 1

Step 2: Apply gg to the result: wherever xx appears in gg, substitute f(x)=2x+1f(x) = 2x + 1 g(f(x))=g(2x+1)=(2x+1)2g(f(x)) = g(2x+1) = (2x+1)^2

Step 3: Expand g(f(x))=4x2+4x+1g(f(x)) = 4x^2 + 4x + 1

Compare:

  • f(g(x))=2x2+1f(g(x)) = 2x^2 + 1
  • g(f(x))=4x2+4x+1g(f(x)) = 4x^2 + 4x + 1

They're DIFFERENT! Order matters!

Evaluating Composite Functions

To find f(g(3))f(g(3)) where f(x)=2x+1f(x) = 2x + 1 and g(x)=x2g(x) = x^2:

Method 1: Find g(x)g(x) first, then ff of that g(3)=32=9g(3) = 3^2 = 9 f(g(3))=f(9)=2(9)+1=19f(g(3)) = f(9) = 2(9) + 1 = 19

Method 2: Use the composite function formula we found f(g(x))=2x2+1f(g(x)) = 2x^2 + 1 f(g(3))=2(3)2+1=18+1=19f(g(3)) = 2(3)^2 + 1 = 18 + 1 = 19

Both methods give the same answer (as they should!).

Part 8: Inverse Functions

What Is an Inverse Function?

An inverse function "undoes" what the original function does.

If ff takes input xx to output yy, then f−1f^{-1} takes input yy back to output xx.

Notation: f−1(x)f^{-1}(x) (read as "ff inverse of xx")

Key property: f(f−1(x))=xf(f^{-1}(x)) = x f−1(f(x))=xf^{-1}(f(x)) = x

When Does an Inverse Exist?

Not every function has an inverse! A function has an inverse ONLY if it's one-to-one (injective):

  • Each output comes from exactly one input
  • No two different inputs produce the same output
  • Passes the horizontal line test: any horizontal line crosses the graph at most once
Example

Does f(x)=2x+3f(x) = 2x + 3 have an inverse?

Check: Different inputs give different outputs

  • f(1)=5f(1) = 5
  • f(2)=7f(2) = 7
  • f(3)=9f(3) = 9
  • ...

Each input xx produces a unique output. YES, it has an inverse. ✓

Does g(x)=x2g(x) = x^2 have an inverse (for all real numbers)?

Check: Multiple inputs give same output

  • g(2)=4g(2) = 4
  • g(−2)=4g(-2) = 4

Same output (4) from different inputs (2 and -2). NO, it doesn't have an inverse for all reals. ✗

(Note: If we restrict the domain to x≥0x \geq 0, then gg is one-to-one and has an inverse.)

Finding Inverse Functions Algebraically

To find f−1(x)f^{-1}(x) from f(x)f(x):

Step 1: Write y=f(x)y = f(x)

Step 2: Swap xx and yy

Step 3: Solve for yy in terms of xx

Step 4: Replace yy with f−1(x)f^{-1}(x)

Example

Find the inverse of f(x)=2x+3f(x) = 2x + 3:

Step 1: Write y=2x+3y = 2x + 3

Step 2: Swap xx and yy x=2y+3x = 2y + 3

Step 3: Solve for yy x−3=2yx - 3 = 2y y=x−32y = \frac{x-3}{2}

Step 4: Write as inverse function f−1(x)=x−32f^{-1}(x) = \frac{x-3}{2}

Verify:

  • f(f−1(x))=f(x−32)=2⋅x−32+3=(x−3)+3=xf(f^{-1}(x)) = f\left(\frac{x-3}{2}\right) = 2 \cdot \frac{x-3}{2} + 3 = (x-3) + 3 = x ✓
  • f−1(f(x))=f−1(2x+3)=(2x+3)−32=2x2=xf^{-1}(f(x)) = f^{-1}(2x+3) = \frac{(2x+3)-3}{2} = \frac{2x}{2} = x ✓
Finding the Inverse of a Linear Function
Example

Find the inverse of f(x)=x3−2f(x) = x^3 - 2:

Step 1: Write y=x3−2y = x^3 - 2

Step 2: Swap xx and yy x=y3−2x = y^3 - 2

Step 3: Solve for yy x+2=y3x + 2 = y^3 y=x+23y = \sqrt[3]{x+2}

Step 4: Write as inverse f−1(x)=x+23f^{-1}(x) = \sqrt[3]{x+2}

Verify: f(f−1(x))=(x+23)3−2=(x+2)−2=xf(f^{-1}(x)) = (\sqrt[3]{x+2})^3 - 2 = (x+2) - 2 = x ✓

Inverse Functions and Composition

The fundamental relationship: f∘f−1=I (identity function)f \circ f^{-1} = I \text{ (identity function)} f−1∘f=If^{-1} \circ f = I

And if you have composite functions: (f∘g)−1=g−1∘f−1(f \circ g)^{-1} = g^{-1} \circ f^{-1}

Note the order reverses when taking inverses of composites!

Example

If f(x)=2x+1f(x) = 2x + 1 and g(x)=x−3g(x) = x - 3, find (f∘g)−1(f \circ g)^{-1}:

Method 1: Find (f∘g)(f \circ g) first, then invert

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

Now invert y=2x−5y = 2x - 5: x=2y−5x = 2y - 5 y=x+52y = \frac{x+5}{2}

(f∘g)−1(x)=x+52(f \circ g)^{-1}(x) = \frac{x+5}{2}

Method 2: Use (f∘g)−1=g−1∘f−1(f \circ g)^{-1} = g^{-1} \circ f^{-1}

Find inverses:

  • f−1(x)=x−12f^{-1}(x) = \frac{x-1}{2}
  • g−1(x)=x+3g^{-1}(x) = x + 3

g−1(f−1(x))=g−1(x−12)=x−12+3=x−1+62=x+52g^{-1}(f^{-1}(x)) = g^{-1}\left(\frac{x-1}{2}\right) = \frac{x-1}{2} + 3 = \frac{x-1+6}{2} = \frac{x+5}{2}

Both methods agree! ✓