Inverse Functions: One-to-One Functions and Their Inverses

Inverse Functions

One-to-One Functions

A one-to-one function is a function in which every element of the domain maps to a unique element in the range. This means that no two different inputs produce the same output. Formally, a function f is one-to-one if for any two different inputs x1 and x2, f(x1) \neq f(x2).

• Key Point: If two different x-values correspond to two different y-values, the function is one-to-one.

• Example: The function f(x) = x is one-to-one because each x-value maps to a unique y-value.

• Counterexample: The function f(x) = x^2 is not one-to-one over all real numbers because both x = 2 and x = -2 map to y = 4.

Horizontal Line Test

The horizontal line test is a graphical method to determine if a function is one-to-one. If every horizontal line intersects the graph of a function at most once, the function is one-to-one.

• Key Point: If a horizontal line crosses the graph more than once, the function is not one-to-one.

• Theorem: A function that is increasing or decreasing on an interval is one-to-one on that interval.

Inverse Functions: Definition and Properties

If y = f(x) is a one-to-one function, the correspondence from the range of f to the domain of f is called the inverse function of f. The symbol f^{-1} is used to denote the inverse function. If y = f(x) is one-to-one, then f has an inverse function f^{-1} such that:

• Domain of f = Range of f-1

• Range of f = Domain of f-1

• f(f^{-1}(x)) = x for x in the domain of f^{-1}

• f^{-1}(f(x)) = x for x in the domain of f

The graph of a one-to-one function f and its inverse f^{-1} are symmetric with respect to the line y = x.

Verifying Inverse Functions

To verify that two functions are inverses, compose them and check if the result is the identity function:

• Example 1: For g(x) = x^3, the inverse is g^{-1}(x) = \sqrt[3]{x}. Verify: $g(g^{-1}(x)) = (\sqrt[3]{x})^3 = x$ and $g^{-1}(g(x)) = \sqrt[3]{x^3} = x$.

• Example 2: For f(x) = 2x + 3, the inverse is f^{-1}(x) = \frac{1}{2}(x - 3). Verify: $f(f^{-1}(x)) = 2 \cdot \frac{1}{2}(x - 3) + 3 = x - 3 + 3 = x$.

• Example 3: For h(x) = 4x - 7, the inverse is h^{-1}(x) = \frac{x + 7}{4}. Verify: $h(h^{-1}(x)) = 4 \cdot \frac{x + 7}{4} - 7 = x + 7 - 7 = x$.

Finding the Inverse of a Function

To find the inverse of a function f(x):

1. Switch x and y in the equation.

2. Solve for y (isolate y on one side).

3. Indicate that the result is f-1(x).

4. Check your answer by verifying $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

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

  1. Write y = 2x + 3.

  2. Switch x and y: x = 2y + 3.

  3. Solve for y: $y = \frac{x - 3}{2}$.

  4. So, f^{-1}(x) = \frac{x - 3}{2}.

Domain-Restricted Inverses

Some functions, such as f(x) = x^2, are not one-to-one over their entire domain. To find an inverse, restrict the domain so the function becomes one-to-one.

• Example: For f(x) = x^2 with x \geq 0, the inverse is f^{-1}(x) = \sqrt{x}.

• Reason for restriction: Without restricting the domain, x^2 is not one-to-one because both positive and negative values of x yield the same y.

Summary Table: Properties of Inverse Functions

Property	Function f	Inverse f-1
Domain	Domain of f	Range of f
Range	Range of f	Domain of f
Composition	$f(f^{-1}(x)) = x$	$f^{-1}(f(x)) = x$
Graph Symmetry	Symmetric about the line $y = x$