Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.


{Use of Tech} ƒ(x) = 1/(x-5)

Verified step by step guidance

1. Step 1: Understand the concept of an inverse function. An inverse function exists if a function is one-to-one (bijective), meaning it passes the horizontal line test. This implies that for every y-value, there is exactly one x-value.
2. Step 2: Analyze the function \( f(x) = \frac{1}{x-5} \). Notice that this is a rational function with a vertical asymptote at \( x = 5 \), where the function is undefined.
3. Step 3: Consider the domain of \( f(x) \). The function is defined for all real numbers except \( x = 5 \). Therefore, the domain is \( (-\infty, 5) \cup (5, \infty) \).
4. Step 4: Check if the function is one-to-one on its domain. Since \( f(x) = \frac{1}{x-5} \) is a strictly decreasing function on each interval \( (-\infty, 5) \) and \( (5, \infty) \), it passes the horizontal line test on these intervals.
5. Step 5: Conclude that the function \( f(x) = \frac{1}{x-5} \) has an inverse on each of the intervals \( (-\infty, 5) \) and \( (5, \infty) \), as it is one-to-one on these intervals.

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(x), its inverse f⁻¹(x) satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f⁻¹. A function has an inverse if it is one-to-one, meaning it passes the horizontal line test, where no horizontal line intersects the graph of the function more than once.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). Understanding the domain and range is crucial for determining where a function is invertible, as the range of the original function becomes the domain of its inverse.

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, the function fails the test and does not have an inverse. This test is particularly useful for visualizing the behavior of functions and identifying intervals where they may be invertible.