Problem 79 - College Algebra

Question

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=\frac{x^{4}}{4} $$

Step-by-Step Solution

Answer

By graphing \( f(x)=\frac{x^{4}}{4} \), examining the graph, and applying the horizontal line test, one can determine whether or not the function is one-to-one, and therefore, whether or not it has an inverse that is a function. If all horizontal lines intersect the function's graph at most once, then the function is one-to-one and does have an inverse that is a function.

Step 1: Graph the Function

Using a graphing utility, plot the points of the function \( f(x)=\frac{x^{4}}{4} \).

Step 2: Examine the Graph

Look at the graph drawn. Does any horizontal line drawn through the graph touch the graph more than once?

Step 3: Apply the Horizontal Line Test

Apply the horizontal test. Draw several horizontal lines through the graph at different heights. If any of these lines intersect the graph more than once, then the function is not one-to-one. If none of the lines intersect the graph more than once, then the function is one-to-one.

Step 4: Determine Whether the Function Has an Inverse That Is a Function

A function has an inverse that is also a function if and only if the original function is one-to-one. So, if the function is found to be one-to-one in the previous step, it does have an inverse that is a function. If not, the inverse of the function will not be a function.

Key Concepts

Graphing Utility

To understand whether a function like \( f(x) = \frac{x^4}{4} \) has an inverse that qualifies as a function, a helpful tool is the graphing utility. Graphing utilities come in the form of both physical calculators and software applications. They allow users to plot mathematical functions quickly and accurately.

When using a graphing utility, follow these simple steps to graph a function:

• Input the function into the utility. Ensure all expressions are entered correctly.
• Adjust the viewing window to capture important features of the graph, such as intercepts and the overall shape.
• Generate the graph to visualize the function. Look out for smooth or continuous patterns.

The graph aids in analyzing the function’s properties, such as growth behavior and symmetry. This visual representation forms the foundation for examining whether or not a function has an inverse that is itself a function. For functions like \( f(x) = \frac{x^4}{4} \), which resembles a parabolic shape, the utility showcases its symmetry — a key point for further investigation.

Horizontal Line Test

The horizontal line test is a simple yet powerful tool used to determine if a function is one-to-one. When you have graphed a function using a graphing utility, this test can help ascertain the possibility of an inverse.
To apply the horizontal line test:

• Draw horizontal lines at various heights across the graph.
• Observe the points where these lines intersect with the graph.

If any horizontal line crosses the graph more than once, then the function is not one-to-one. This indicates that multiple values of \( x \) yield the same \( f(x) \), ruling out the chance of the function having an inverse that is also a function.
For \( f(x) = \frac{x^4}{4} \), drawing such lines will reveal intersections that occur more than once. This shows it's not one-to-one since the graph is symmetric about the y-axis, typical for even-powered functions.

One-to-One Function

The term 'one-to-one' refers to a function where each output value is produced by exactly one input value. This property is crucial when considering the existence of an inverse function that is also a function.
Understanding if a function is one-to-one involves looking back at the horizontal line test. A one-to-one function will not have any horizontal line intersecting its graph more than once, ensuring its inverse is also a function.
For the function \( f(x) = \frac{x^4}{4} \), performing the horizontal line test shows that it fails to pass this criterion. Why is this important? Because it means that there are some \( f(x) \) values achieved by more than one \( x \). This inability to meet the one-to-one requirement confirms that while an inverse may still exist, it won't satisfy the condition needed to also be a function.
Hence, not all functions can have their inverses immediately considered functions. For one-to-one functions, however, they pair neatly with their inverses, granting a full functional cycle.