Inverse Functions

What is an Inverse?

An inverse function undoes what the original function does.

If ff takes aa to bb, then f−1f^{-1} takes bb back to aa.

f(a)=b⟺f−1(b)=af(a) = b \quad \Longleftrightarrow \quad f^{-1}(b) = a

The Defining Property

A function f−1f^{-1} is the inverse of ff if:

f(f−1(x))=xandf−1(f(x))=xf(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x

Applying a function and its inverse (in either order) gets you back where you started.

Example

Let f(x)=2x+3f(x) = 2x + 3.

If we claim f−1(x)=x−32f^{-1}(x) = \dfrac{x - 3}{2}, let’s verify:

Check f(f−1(x))f(f^{-1}(x)):

f(f−1(x))=f(x−32)=2⋅x−32+3=(x−3)+3=x✓\begin{aligned} f(f^{-1}(x)) &= f\left(\frac{x - 3}{2}\right) \\[0.5em] &= 2 \cdot \frac{x - 3}{2} + 3 \\[0.5em] &= (x - 3) + 3 = x \quad \checkmark \end{aligned}

Check f−1(f(x))f^{-1}(f(x)):

f−1(f(x))=f−1(2x+3)=(2x+3)−32=2x2=x✓\begin{aligned} f^{-1}(f(x)) &= f^{-1}(2x + 3) \\[0.5em] &= \frac{(2x + 3) - 3}{2} \\[0.5em] &= \frac{2x}{2} = x \quad \checkmark \end{aligned}

Finding an Inverse

Steps:

  1. Write y=f(x)y = f(x)
  2. Swap xx and yy
  3. Solve for yy
  4. The result is f−1(x)f^{-1}(x)

Example: Find the inverse of f(x)=3x−5f(x) = 3x - 5.

Step 1: y=3x−5y = 3x - 5

Step 2: x=3y−5x = 3y - 5

Step 3: Solve for yy

x+5=3yy=x+53\begin{aligned} x + 5 &= 3y \\ y &= \frac{x + 5}{3} \end{aligned}

Result: f−1(x)=x+53f^{-1}(x) = \dfrac{x + 5}{3}

Example: Find the inverse of f(x)=x3+2f(x) = x^3 + 2.

Step 1: y=x3+2y = x^3 + 2

Step 2: x=y3+2x = y^3 + 2

Step 3: Solve for yy

x−2=y3y=x−23\begin{aligned} x - 2 &= y^3 \\ y &= \sqrt[3]{x - 2} \end{aligned}

Result: f−1(x)=x−23f^{-1}(x) = \sqrt[3]{x - 2}

Not All Functions Have Inverses

For a function to have an inverse, it must be one-to-one: each output comes from exactly one input.

One-to-one: f(x)=2x+1f(x) = 2x + 1 ✓ (every y-value comes from one x-value)

Not one-to-one: f(x)=x2f(x) = x^2 ✗ (both x=2x = 2 and x=−2x = -2 give y=4y = 4)

The Horizontal Line Test

A function has an inverse if and only if every horizontal line crosses its graph at most once.

  • f(x)=x3f(x) = x^3 passes → has an inverse
  • f(x)=x2f(x) = x^2 fails → no inverse (unless we restrict the domain)

Restricting the Domain

We can create an inverse by limiting the domain.

f(x)=x2f(x) = x^2 has no inverse on all real numbers.

But f(x)=x2f(x) = x^2 for x≥0x \geq 0 does have an inverse: f−1(x)=xf^{-1}(x) = \sqrt{x}

We restricted to the right half of the parabola, making it one-to-one.

Graphs of Inverse Functions

The graph of f−1f^{-1} is the reflection of the graph of ff across the line y=xy = x.

This makes sense: if (a,b)(a, b) is on the graph of ff, then (b,a)(b, a) is on the graph of f−1f^{-1}.

Domain and Range Swap

For ff and f−1f^{-1}:

  • Domain of ff = Range of f−1f^{-1}
  • Range of ff = Domain of f−1f^{-1}

Example: f(x)=xf(x) = \sqrt{x} has domain [0,∞)[0, \infty) and range [0,∞)[0, \infty).

Its inverse f−1(x)=x2f^{-1}(x) = x^2 (for x≥0x \geq 0) has domain [0,∞)[0, \infty) and range [0,∞)[0, \infty).