Problem 94 - Precalculus Student Solutions Manual 5th

Question

Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse. $$\begin{array}{l} f(x)=x^{4}-5 x^{2} \\ {[-3,3] \text { by }[-8,8]} \end{array}$$

Step-by-Step Solution

Answer

The function \( f(x) = x^4 - 5x^2 \) is not one-to-one.

Step 1 - Enter the function

On the graphing calculator, input the function as: \( f(x) = x^4 - 5x^2 \).

Step 2 - Set the viewing window

Adjust the graphing calculator's viewing window settings to: X-axis: \[-3, 3\] Y-axis: \[-8, 8\].

Step 3 - Graph the function

Plot the function on the calculator to visualize the graph.

Step 4 - Determine if the function is one-to-one

Analyze the graph to see if it passes the Horizontal Line Test. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.

Step 5 - Conclusion on one-to-one property

From observing the graph, it is clear that \( f(x) = x^4 - 5x^2 \) does not pass the Horizontal Line Test, as some horizontal lines intersect the function at more than one point.

Key Concepts

one-to-one functions

A one-to-one function is a function where each input (x-value) corresponds to exactly one output (y-value), and each output also corresponds to exactly one input. To put it simply, no two different inputs map to the same output. This is an important concept in mathematics as it ensures the function has an inverse that is also a function. For instance, if you have a function, say, g(x), and you know it is one-to-one, then you can find another function, g^-1(x), which essentially 'reverses' the effect of g(x). This concept of having a reverse or 'undo' function is crucial in many mathematical applications. Keep in mind, not all functions are one-to-one.

horizontal line test

The horizontal line test is a simple yet powerful tool used to determine if a function is one-to-one. To use this test, you imagine drawing horizontal lines at different points along the y-axis of your graph. If at any position, a horizontal line intersects the graph of the function at more than one point, it means that the function is not one-to-one. Why does this work? Because if a horizontal line crosses the graph at multiple points, it means that there are different x-values (inputs) that produce the same y-value (output). This fails the definition of a one-to-one function. In our example, the function f(x) = x⁴ - 5x² did not pass the horizontal line test as some horizontal lines intersect the graph at more than one point.

inverse functions

Inverse functions are a fascinating concept. If you have a one-to-one function, it means you can find an inverse function that exactly reverses its effect. Symbolically, if you have a function f(x), then its inverse is denoted as f^-1(x). To understand this further, consider the function f(x) that converts Celsius to Fahrenheit. Its inverse would convert Fahrenheit back to Celsius. Finding the inverse involves switching the x and y roles and then solving for y. However, only one-to-one functions can have inverses that are also functions. In our given problem, since f(x) = x⁴ - 5x² is not one-to-one, it does not have an inverse function.