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College Algebra Unit 7: Inverse functions

An inverse function undoes the original: if f sends a to b, then f⁻¹ sends b back to a.

Covers: One-to-one test, finding inverses, graphs of inverses

Study guide

A function is one-to-one if each output comes from exactly one input. Graphically, it passes the horizontal line test: no horizontal line crosses the graph more than once. Only one-to-one functions have inverse functions. Functions that aren't one-to-one, like x², can have their domains restricted (x ≥ 0) to make them invertible.

The inverse f⁻¹ undoes f: if f(a) = b, then f⁻¹(b) = a. To find it algebraically, replace f(x) with y, swap x and y, and solve for y. For f(x) = 2x + 3, swapping gives x = 2y + 3, so f⁻¹(x) = (x − 3)/2. The notation f⁻¹ means the inverse function, not 1/f(x).

Verify an inverse by composing: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

The graph of f⁻¹ is the reflection of the graph of f across the line y = x, because swapping coordinates (a, b) → (b, a) is that reflection. You can sketch an inverse by swapping the coordinates of key points. Exponential and logarithmic functions are an important inverse pair, as are squaring (restricted to x ≥ 0) and square roots.

In context, f⁻¹(b) is the input that produces the output b, so its units are the original input units. For example, if f(t) is a temperature at time t, then f⁻¹(70) is the time when the temperature is 70. To invert f(x) = (ax + b)/(cx + d), swap x and y, multiply both sides by the denominator, collect the y-terms on one side, factor out y, and divide.

Key ideas

• Only one-to-one functions have inverses. Use the horizontal line test.
• To find an inverse: write y = f(x), swap x and y, solve for y.
• The graph of f⁻¹ is the reflection of f over the line y = x.
• Domain of f becomes range of f⁻¹, and vice versa.
• Check: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Formulas and rules

Common mistakes

• Writing f⁻¹(x) = 1/f(x). The −1 is not an exponent.
• Forgetting to restrict the domain, e.g. for f(x) = x².

Words to know

Horizontal line test

A function is one-to-one (has an inverse) if no horizontal line hits its graph twice.

Finding an inverse function

Swap x and y, then solve for y.

One-to-one function

Each output comes from exactly one input, so it has an inverse.

Inverse function graph

The reflection of f over the line y = x.

f⁻¹(x) notation

The inverse function, not 1/f(x).

Inverse function (f⁻¹)

Undoes f: if f(a) = b then f⁻¹(b) = a.

Verifying inverses

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Domain and range of f⁻¹

Domain of f⁻¹ = range of f, and vice versa.

Restricting a domain

Limiting inputs (like x ≥ 0 for x²) to make a function one-to-one.

Sample practice questions

17 more questions like these, from level 1 to 5, are in the practice set.