What is an inverse function?

An inverse function “undoes” what the original function does.

• If f(x) takes an input x and gives output y, the inverse function f⁻¹(x) takes y and gives back x.

• Think of it like reversing a process:

• Notation: f⁻¹(x) (read as “f inverse of x”)

Step 1: Check if a function has an inverse

• A function must be one-to-one to have an inverse.

  • One-to-one means each output corresponds to exactly one input.

• Test using the Horizontal Line Test:

  • Draw horizontal lines on the graph of f(x).

  • If any horizontal line crosses the graph more than once, it does not have an inverse.

Step 2: How to find the inverse function

Steps:

1. Start with y = f(x)

2. Swap x and y

3. Solve for y

4. Replace y with f⁻¹(x)

Example 1:

Step 1: Write y instead of f(x)

Step 2: Swap x and y

Step 3: Solve for y

Step 4: Write as inverse function

✅ Check:

• f(2) = 2*2 + 3 = 7

• f⁻¹(7) = (7 - 3)/2 = 2 ✔

Step 3: Graph of inverse function

• The graph of f⁻¹(x) is a reflection of f(x) across the line y = x.

• This makes sense because the inverse switches x and y.

Tip: Always check if the inverse makes sense visually!

Step 4: Domain and range

• The domain of f(x) becomes the range of f⁻¹(x)

• The range of f(x) becomes the domain of f⁻¹(x)

Example:

Step 5: Inverse of more complex functions

Example 2:

Step 1: Write y = f(x)

Step 2: Swap x and y

Step 3: Solve for y

Step 4: Write as inverse function

✅ Check: f(f⁻¹(1)) = 1 ✔

Why inverse functions matter

• Solve equations: f(x) = a → x = f⁻¹(a)

• Undo transformations

• Used in science, engineering, and real-world problems like:

  • Converting temperatures: Celsius ↔ Fahrenheit

  • Converting currencies

  • Undoing scaling or shifting in graphs

Common beginner mistakes

1. ❌ Forgetting to swap x and y

2. ❌ Not solving for y correctly

3. ❌ Using a function that is not one-to-one

4. ❌ Ignoring domain and range restrictions