Problem 22 - College Algebra

Question

For the following exercises, use a graphing utility to determine whether each function is one-to-one. $$ f(x)=x^{3}-27 $$

Step-by-Step Solution

Answer

The function \(f(x) = x^3 - 27\) is one-to-one.

Step 1: Understanding a One-to-One Function

A function is one-to-one if and only if no two different inputs, say \(x_1\) and \(x_2\), in the function's domain correspond to the same output. In other words, \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\).

Step 2: Graphical Test for One-to-One

One effective method to test if a function is one-to-one is by using the Horizontal Line Test on its graph. If every horizontal line intersects the graph of the function at most once, the function is one-to-one.

Step 3: Graph the Function

Enter the function \(f(x) = x^3 - 27\) into a graphing utility. Observe the shape and direction of the graph. This function is cubic, which usually appears as a smooth, continuous curve with no repeated y-values.

Step 4: Apply the Horizontal Line Test

Plot a few horizontal lines across the graph. Observe whether any line intersects the graph more than once. For the function \(f(x) = x^3 - 27\), each horizontal line intersects it exactly once, confirming that it passes the Horizontal Line Test.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical functions. It can be a software application, an online tool, or even a dedicated calculator. Graphing utilities allow you to input equations and instantly see their graphs.

• They help in understanding complex functions by providing a visual representation.
• With graphing utilities, you can easily view characteristics such as intercepts, slopes, and behavior of functions.
• They are invaluable in conducting tests like the Horizontal Line Test to check if a function is one-to-one.

For example, when you enter the function \(f(x) = x^3 - 27\) into a graphing utility, you'll see its curve, making it easier to apply tests and analyze its properties.

Horizontal Line Test

The Horizontal Line Test is a simple visual method used to determine if a function is one-to-one. To conduct this test:

• Draw several horizontal lines across the graph of the function.
• Observe where these lines intersect the graph.

If at any point a horizontal line crosses the graph more than once, the function is not one-to-one. However, if every horizontal line intersects the graph at most once, then the function is indeed one-to-one.
For the function \(f(x) = x^3 - 27\), using a graphing utility, you can quickly draw horizontal lines and see that they each intersect the graph only once. This confirms that the function passes the Horizontal Line Test, indicating it is one-to-one.

Cubic Function

A cubic function is a polynomial function of degree three, and it can generally be expressed in the form \(f(x) = ax^3 + bx^2 + cx + d\). These functions are characterized by their distinct graphs:

• They have a continuous and smooth structure with no breaks.
• Their basic shape resembles an 'S' or backward 'S', depending on the leading coefficient's sign.

Cubic functions can have one or two turns, creating a local maximum and minimum. However, the function \(f(x) = x^3 - 27\) is a straightforward cubic function with no quadratic or linear components. This specific function shifts the basic cubic graph downward by 27 units, due to the constant term -27. Observing the graph of this function using a graphing utility helps in understanding its characteristics and confirming its behavior as a one-to-one function through tests like the Horizontal Line Test.