Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$f(x)=x^{5}-7$$

Step-by-Step Solution

Answer

The function \(f(x) = x^{5}-7\) is one-to-one, therefore, it has an inverse.

Step 1: Graph the function

Use a graphing utility to plot the function \(f(x) = x^{5}-7\). The graph will be a curve that, for increasing values of \(x\), also increase in value (and decrease for decreasing values of \(x\)). There will be no repeated values of \(f(x)\) for different values of \(x\).

Step 2: Perform the horizontal line test

To check whether the function is one-to-one, a horizontal line is drawn. For the function to be one-to-one, the horizontal line should intersect the graph at only one point for all possible lines. If we look at the graph of the function \(f(x) = x^{5}-7\), we can see that any horizontal line will intercept the function only once.

Step 3: Evaluate the result

Since every horizontal line only intersects the graph of \(f(x) = x^{5}-7\) at one point, we can conclude that this function is one-to-one. Therefore, it has an inverse function

Key Concepts

Horizontal Line Test

The Horizontal Line Test is a simple visual method to determine if a function is one-to-one. A function is considered one-to-one if any horizontal line drawn across the graph of the function cuts the graph at most once. To apply this test, imagine drawing a series of horizontal lines at different heights across the graph of the function. If you find that none of these lines intersect the graph more than once, then the function passes the test.

- **Why is this important?**
Passing the Horizontal Line Test implies that the function is one-to-one, meaning each output corresponds to only one input. This characteristic is crucial in determining whether the function has an inverse.
- **Application to our function:**
For the given function \(f(x) = x^5 - 7\), when we plot the graph, any horizontal line we draw will intersect this curve at only one point. Thus, \(f(x) = x^5 - 7\) is one-to-one.

Inverse Functions

An inverse function essentially reverses the operation of the original function. In simpler terms, if a function \(f\) turns \(x\) into \(y\), then its inverse takes \(y\) back to \(x\). Not all functions have inverses.

- **Necessary condition:**
A function must be one-to-one to have an inverse. Why? Because if two different input values can produce the same output, there's no single way to revert from output to input.
- **Finding an inverse:**
If you confirm, through the Horizontal Line Test, that a function is one-to-one, you can find its inverse by switching the roles of \(x\) and \(y\) in the equation and solving for \(x\).
- **Example with \(f(x) = x^5 - 7\):**
The function passes the Horizontal Line Test, indicating an inverse exists. To find the inverse, express \(x\) in terms of \(y\), giving \(x = (y + 7)^{1/5}\). Hence, \(f^{-1}(x) = (x + 7)^{1/5}\).

Graphing Functions

Graphing a function provides a visual representation and can reveal characteristics such as symmetry, intercepts, and behavior at infinity.

- **Why graph functions?**
Graphs allow us to visually analyze the function and perform further tests, like the Horizontal Line Test, making it easier to understand the function's properties.
- **Using a graphing utility:**
For functions like \(f(x) = x^5 - 7\), a graphing calculator or software can help quickly plot the graph.
- **Interpreting the graph of \(f(x) = x^5 - 7\):**
The curve starts in the lower left, climbs steadily toward the upper right, showing a smooth increase without plateaus or dips that might indicate repeating outputs. Therefore, smooth and consistent behavior confirms the function's one-to-one nature. This also visually supports our understanding of the inverse function's existence.