Problem 61 - 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. $$y=\frac{1}{x}$$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \frac{1}{x} \); domain and range for each are \( x \neq 0 \) and \( y \neq 0 \).

Step 1: Determine if the function is One-to-One

To check if the function is one-to-one, we can use the horizontal line test or check that every y-value of the function corresponds to exactly one x-value. The function \( y = \frac{1}{x} \) is one-to-one because for each output \( y \), there is a unique input \( x \).

Step 2: Find the Inverse Function

To find the inverse function, start with the equation \( y = \frac{1}{x} \). Swap \( x \) and \( y \): \( x = \frac{1}{y} \). Solve for \( y \): Multiply both sides by \( y \) to get \( xy = 1 \), then divide by \( x \) which gives \( y = \frac{1}{x} \). Therefore, the inverse function is \( f^{-1}(x) = \frac{1}{x} \).

Step 3: Domain and Range of the Functions

For the function \( f(x) = \frac{1}{x} \), the domain is all real numbers except zero, \( x eq 0 \). Similarly, the range is all real numbers except zero, \( y eq 0 \). The inverse function \( f^{-1}(x) = \frac{1}{x} \) has the same domain and range: all real numbers except zero, since the same expression \( \frac{1}{x} \) is used in both functions.

Step 4: Graph the Functions

Graph the function \( y = \frac{1}{x} \) and its inverse \( y = \frac{1}{x} \) on the same axes. Both graphs will be hyperbolas, symmetric with respect to the line \( y = x \). Since \( f \) and \( f^{-1} \) are the same function here, they overlap perfectly.

Key Concepts

One-to-One Functions

A one-to-one function is a function where each input corresponds to exactly one output, and each output corresponds to exactly one input. This property ensures that the function has an inverse.

• In a one-to-one function, no two different values of the independent variable (usually represented as \(x\)) will yield the same dependent variable (\(y\)).
• This means that if \(f(a) = f(b)\), then \(a = b\) for a one-to-one function.

For example, the function \( y = \frac{1}{x} \) is one-to-one because for each unique \(x\), there is a unique \(y\), and vice versa. This property was verified in the original solution by recognizing that every \(y\)-value corresponds to a unique \(x\)-value.

Horizontal Line Test

The Horizontal Line Test is a simple visual tool used to determine if a function is one-to-one.

• By drawing horizontal lines across the graph of a function, we check if any of these lines intersect the graph at more than one point.
• If a horizontal line intersects the graph in more than one place, the function fails the test and is not one-to-one.
• If no horizontal line can intersect the graph more than once, the function is one-to-one!

Consider the function from our exercise, \(y=\frac{1}{x}\). The horizontal line test can be applied here since at any position, these lines will touch the graph of the function only once, hence confirming it’s one-to-one.

Domain and Range

Understanding domain and range is crucial when dealing with functions and their inverses.

• The domain of a function is the set of all possible input values (\(x\)-values).
• The range is the set of all possible output values (\(y\)-values).

In our given problem, for the function \( f(x) = \frac{1}{x} \), the domain is all real numbers except zero, since division by zero is undefined. Similarly, the range is also all real numbers except zero.
When we have a function and its inverse, the domain of the original function becomes the range of the inverse function and vice-versa. For the inverse \( f^{-1}(x) = \frac{1}{x} \), the domain and range similarly exclude zero.

Graphing Inverse Functions

Graphing inverse functions involves plotting both the function and its inverse, often showing their reflection across the line \(y=x\).

• For the given function \(y = \frac{1}{x}\), we found that its inverse is the same, \(f^{-1}(x) = \frac{1}{x}\).
• These functions are a perfect mirror image of each other across the line \(y = x\).

When graphing, observe that both the function and the inverse are hyperbolas and they overlap completely because they are identical. Their symmetric nature relative to the line \(y=x\) is a clear visual implication of them being inverses. This overlapping on the graph confirms their symmetric properties inherent in inverse functions.