Problem 64 - A Graphical Approach to Precalculus with Limits

Question

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) 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. $$y=\frac{3}{x-4}$$

Step-by-Step Solution

Answer

\( f^{-1}(x) = \frac{3+4x}{x} \); domain: \( x \neq 0 \), range: \( y \neq 4 \).

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

To determine if the function is one-to-one, check if each output value of the function corresponds to exactly one input value. For rational functions like \( y = \frac{3}{x-4} \), the function is one-to-one because the inverse operation (multiplying by \( x-4 \)) is also a function.

Step 2: Solve for the inverse function

To find the inverse, we start by replacing \( y \) with \( x \), giving \( x = \frac{3}{y-4} \). To solve for \( y \), multiply both sides by \( y-4 \), giving \( x(y-4) = 3 \). Distribute \( x \) to get \( xy - 4x = 3 \). Next, solve for \( y \) by adding \( 4x \) to both sides to get \( xy = 3 + 4x \), then divide by \( x \) to isolate \( y \), resulting in \( y = \frac{3 + 4x}{x} \).

Step 3: Write the inverse function

The inverse function of \( f(x) \) in terms of \( y \) is \( f^{-1}(x) = \frac{3+4x}{x} \). Now, the inverse function is properly written and formatted.

Step 4: Determine the domain and range

For the original function \( f(x) = \frac{3}{x-4} \), the domain is \( x eq 4 \) and the range is \( y eq 0 \). For the inverse function \( f^{-1}(x) = \frac{3+4x}{x} \), the domain is \( x eq 0 \) and the range is \( y eq 4 \).

Step 5: Graph the functions

To graph \( f(x) = \frac{3}{x-4} \) and \( f^{-1}(x) = \frac{3+4x}{x} \), plot both on the same axes. The function \( f(x) \) will be a hyperbola with a vertical asymptote at \( x = 4 \) and a horizontal asymptote at \( y = 0 \). The inverse \( f^{-1}(x) \) is also a hyperbola, but with a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 4 \). Both graphs should be symmetric with respect to the line \( y = x \).

Key Concepts

One-to-One Functions

Understanding whether a function is one-to-one is crucial for finding its inverse. A function is one-to-one if each output value is the result of exactly one input value. This means that no two different inputs should produce the same output. For the function \( y = \frac{3}{x-4} \), every input \( x \) (other than \( x = 4 \), where the function is undefined) leads to a unique output value. Thus, this function is one-to-one.

Being able to determine if a function is one-to-one helps decide if an inverse can exist. If a function is one-to-one, you can confidently find and work with its inverse, knowing the relationship is consistently reversible.

Domain and Range

The domain and range of a function tell us the possible inputs and outputs, respectively. For the original function \( f(x) = \frac{3}{x-4} \), the domain excludes \( x = 4 \) because division by zero is undefined. So, the domain is all real numbers except 4. The range is all real numbers except 0. This is because the output of the function (\( y \)) never reaches zero as \( x \) approaches 4, due to the nature of the division.

When exploring the inverse function \( f^{-1}(x) = \frac{3+4x}{x} \), the domain now excludes \( x = 0 \) since this would also lead to division by zero. The range in this case is all real numbers except 4, as the value never becomes 4 due to the asymptotic behavior. Understanding these sets of inputs and outputs for both the function and its inverse is critical for graphing and real-world applications.

Graphing Functions

Graphing functions helps visually understand their behavior and properties, like symmetries and asymptotes. The graph of \( f(x) = \frac{3}{x-4} \) is a hyperbola with a vertical asymptote at \( x = 4 \) and a horizontal asymptote at \( y = 0 \). This graph means that as \( x \) approaches 4 from either side, \( y \) increases or decreases sharply, and \( y \) approaches 0 but never reaches it.

For the inverse function \( f^{-1}(x) = \frac{3+4x}{x} \), the hyperbola has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 4 \). These asymptotes indicate where the function's output approaches infinity or a fixed point, influencing how the function can be used in different contexts. Graphing both \( f \) and \( f^{-1} \) on the same axes can reveal the line of symmetry \( y = x \), showing the mirroring property of the inverse relationship and solidifying the concept of inverse functions.