The domain of an inverse function, f-1(x), is directly related to the range of the original function, f(x). Finding it is often straightforward if you know the range of the original function.
Understanding the Relationship
The key concept is that the input and output values switch places when you go from a function to its inverse. This means:
- Domain of f-1(x) = Range of f(x)
- Range of f-1(x) = Domain of f(x)
This is a fundamental principle explained in the reference: "The outputs of the function f are the inputs to f−1, so the range of f is also the domain of f−1. Likewise, because the inputs to f are the outputs of f−1, the domain of f is the range of f−1."
Steps to Find the Domain of f-1(x)
Here's a step-by-step approach:
- Find the Range of the Original Function, f(x): This is often the most challenging part. Techniques for finding the range depend on the type of function:
- Polynomials: Even-degree polynomials (like quadratics) have a restricted range based on their vertex. Odd-degree polynomials often have a range of all real numbers.
- Rational Functions: Consider horizontal asymptotes and any values excluded due to vertical asymptotes.
- Exponential Functions: Typically have a range of (0, ∞) or (-∞, 0) depending on the base and any vertical shifts.
- Logarithmic Functions: The range is usually all real numbers.
- Radical Functions: Consider the domain and how the function transforms those values. For example, √x has a range of [0, ∞).
- Trigonometric Functions: Know the standard ranges (e.g., sine and cosine have a range of [-1, 1]).
- The Range of f(x) is the Domain of f-1(x): Once you've determined the range of f(x), you've also found the domain of f-1(x). Simply state the range of f(x) as the domain of f-1(x).
Examples
Let's illustrate this with examples:
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Example 1: Suppose f(x) = x2 for x ≥ 0. The range of f(x) is [0, ∞). Therefore, the domain of f-1(x) is [0, ∞).
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Example 2: Suppose f(x) = ex. The range of f(x) is (0, ∞). Therefore, the domain of f-1(x) is (0, ∞). Note that f-1(x) = ln(x), and the domain of the natural logarithm function is indeed (0, ∞).
Table Summarizing the Relationship
| Feature | f(x) | f-1(x) |
|---|---|---|
| Domain | Set of inputs | Range of f(x) |
| Range | Set of outputs | Domain of f(x) |
