An inverse function reverses the action of another function. If a function takes an input x and produces an output y, its inverse takes that y and returns the original x. This idea matters because it lets us undo operations in algebra, solve equations, and interpret relationships in physics, chemistry, and data analysis.
On a graph, a function and its inverse appear as mirror images across the line y = x.
To find an inverse algebraically, write y = f(x), swap x and y, then solve for y. The swap reflects the fact that inputs and outputs trade roles. A function has an inverse that is also a function only when each output comes from exactly one input, which is checked with the horizontal line test.
To verify two functions are inverses, compose them both ways and check that f(g(x)) = x and g(f(x)) = x on the allowed domains.
Key Facts
- If f(a) = b, then f⁻¹(b) = a.
- The graph of f⁻¹(x) is the reflection of the graph of f(x) across y = x.
- To find an inverse: write y = f(x), swap x and y, then solve for y.
- A function has an inverse function only if it passes the horizontal line test.
- Inverse verification requires f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
- For f(x) = mx + b with m ≠ 0, f⁻¹(x) = (x - b)/m.
Vocabulary
- Inverse function
- A function that reverses another function by mapping each output back to its original input.
- Composition
- The process of using the output of one function as the input of another, written as f(g(x)).
- Horizontal line test
- A graph test that checks whether any horizontal line crosses a graph more than once.
- One-to-one function
- A function in which no two different inputs produce the same output.
- Domain restriction
- A limit placed on the possible input values of a function so that an inverse may exist as a function.
Common Mistakes to Avoid
- Forgetting to swap x and y is wrong because the inverse is found by reversing the roles of input and output.
- Assuming every function has an inverse function is wrong because functions that fail the horizontal line test do not have inverses that are functions without restricting the domain.
- Reflecting across the x-axis or y-axis is wrong because inverse graphs are reflected across the line y = x.
- Checking only f(g(x)) = x is incomplete because true inverse functions must satisfy both f(g(x)) = x and g(f(x)) = x on the correct domains.
Practice Questions
- 1 Find the inverse of f(x) = 3x - 12, then verify your answer by composition.
- 2 Find the inverse of f(x) = (x + 5)/2 and evaluate f⁻¹(9).
- 3 The graph of y = x² fails the horizontal line test on all real numbers. Explain how restricting its domain to x ≥ 0 changes whether it has an inverse function.