Problem 62 - A Graphical Approach to College Algebra

Question

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{4}{x}$$

Step-by-Step Solution

Answer

The function is one-to-one, and its inverse is the same: \(f^{-1}(x) = \frac{4}{x}\). Domain and range for both are \((-\infty, 0) \cup (0, \infty)\).

Step 1: Check if the function is one-to-one

A function is one-to-one if each value in the range corresponds to exactly one value in the domain. To verify this, use the horizontal line test. The graph of the function \(y = \frac{4}{x}\) shows that each horizontal line intersects the curve at most once, confirming it is one-to-one.

Step 2: Write the inverse function

To find the inverse function, replace \(y\) with \(x\) and \(x\) with \(y\) in the equation \(y = \frac{4}{x}\). This gives \(x = \frac{4}{y}\). Solve for \(y\) by multiplying both sides by \(y\) and dividing by \(x\), resulting in \(y = \frac{4}{x}\). Thus, the inverse function is \(f^{-1}(x) = \frac{4}{x}\).

Step 3: Graph the functions

Graph both \(f(x) = \frac{4}{x}\) and its inverse \(f^{-1}(x) = \frac{4}{x}\) on the same set of axes. The two graphs are identical, since the function is its own inverse. Both graphs are hyperbolas, symmetric with respect to the line \(y = x\).

Step 4: Provide domain and range of f and f^{-1}

For \(f(x) = \frac{4}{x}\), the domain excludes \(x = 0\), so it is \(x \in (-\infty, 0) \cup (0, \infty)\). The range is the same, \(y \in (-\infty, 0) \cup (0, \infty)\). For \(f^{-1}(x) = \frac{4}{x}\), the domain and range are identical to those of \(f\).

Step 5: Verify the inverse property

Check that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). For both cases, substituting back results in the identity function that equals \(x\), confirming \(f(x)\) is indeed its own inverse.

Key Concepts

One-to-One Functions

One-to-one functions are a unique kind of function where every element in the domain is paired with a distinct element in the range. This means no two different inputs map to the same output. To verify if a function is one-to-one, you can use the horizontal line test. Simply put, if any horizontal line cuts the function's graph at more than one point, then it is not one-to-one.

In the case of the function \( y = \frac{4}{x} \), applying the horizontal line test reveals that each horizontal line intersects the curve at most once. So, each \( x \) maps to a unique \( y \), confirming it is one-to-one. Knowing a function is one-to-one is crucial when finding its inverse, as only one-to-one functions have inverses across their entire domains.

Domain and Range

Understanding the domain and range is pivotal when dealing with functions and their inverses. The domain of a function consists of all possible input values, while the range contains all possible output values.

For the function \( f(x) = \frac{4}{x} \), the domain excludes \( x = 0 \), since division by zero is undefined. Thus, the domain is \( x \in (-\infty, 0) \cup (0, \infty) \). This means the function accepts any real number except zero. Similarly, the range of the function is \( y \in (-\infty, 0) \cup (0, \infty) \), as it can produce any real number except zero. For its inverse \( f^{-1}(x) = \frac{4}{x} \), both the domain and range remain the same, maintaining the exclusion of zero.

Graphing Functions

Graphing is an effective way to visually interpret and understand functions. For the function \( f(x) = \frac{4}{x} \) and its inverse \( f^{-1}(x) = \frac{4}{x} \), plotting these graphs involves illustrating a hyperbola.

Both functions are symmetric with respect to the line \( y = x \). This symmetry signifies that for any point \((a, b)\) on the graph of \( f(x) \), the point \((b, a)\) will lie on the graph of \( f^{-1}(x) \). Since both are identical, you graph them simultaneously, recognizing that they reflect each other over the line \( y = x \). This geometric insight confirms the inverse relationship, showcasing the consistency of one-to-one nature and validates the inverse calculation process.