The domain of f is the range of f⁻¹, and the domain of f⁻¹ is the range of f
What are inverse functions
Inverse functions are pairs of functions that undo each other.
If you have a function f, its inverse, denoted as f⁻¹, is a function that undoes the effect of f.
Hence we can say that
- the input (domain) of f, is the output (range) of f⁻¹ and
- the input of f⁻¹, is the output of f
1. **Domain and Range Interchange:** If \( (a, b) \) is a point on the graph of \( f \), then \( (b, a) \) is a point on the graph of \( f^{-1} \). In other words, the domain of \( f \) becomes the range of \( f^{-1} \), and vice versa.
2. **Composition:** The composition of a function and its inverse is the identity function. That is, \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1} \), and \( f^{-1}(f(x)) = x \) for all \( x \) in the domain of \( f \).
3. **Notation:** \( f^{-1}(x) \) is read as "f-inverse of x" and is not the same as \( \frac{1}{f(x)} \).
4. **Horizontal Line Test:** A function \( f \) has an inverse if and only if every horizontal line intersects the graph of \( f \) at most once.
It's important to note that not all functions have inverses. For a function to have an inverse, it must satisfy the horizontal line test, meaning that no horizontal line intersects the graph at more than one point.
For example, if \( f(x) = 2x + 3 \), then its inverse function \( f^{-1}(x) \) is \( f^{-1}(x) = \frac{x - 3}{2} \), and the composition \( f(f^{-1}(x)) = x \) holds true.
Inverse functions play a crucial role in solving equations, finding roots, and understanding the relationships between input and output in various mathematical contexts.