Problem 79 - College Algebra Essentials

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

The function \(f(x)=x^{4}/4\) does not have an inverse that is a function because it does not pass the horizontal line test, hence it's not a one-to-one function.

Step 1: Graphing the Function

First, we are going to graph the function \(f(x)=x^{4}/4\). The function is a simple polynomial divided by a constant, so it can be graphed by plotting a set of points or using a graphing tool.

Step 2: Use the Horizontal Line Test

Now, we need to apply the horizontal line test. This test involves drawing lines horizontally across your graph and checking if any line cuts the graph at more than one point. If a line does, then the function is not one-to-one, and consequently, it will not have an inverse that is a function.

Step 3: Analyzing the Graph

From the graph, it can be observed that any drawn horizontal line will intersect the graph at more than one point, implying that the function is not one-to-one. Hence, the function does not have an inverse that is a function.