Problem 11 - Algebra 2 and Trigonometry

Question

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(0,8),(1,7),(2,6),(3,5),(4,4)\\} $$

Step-by-Step Solution

Answer

The function has an inverse: \{(8,0), (7,1), (6,2), (5,3), (4,4)\}.

Step 1: Understand the Criteria for Inverses

A function has an inverse if and only if it is a one-to-one function, meaning each element of the domain (x-values) maps to a unique element in the range (y-values) and vice versa. This is called the Horizontal Line Test: no horizontal line should intersect the graph of the function more than once.

Step 2: Check for One-to-One Mapping

Review the given function pairs: \((0,8), (1,7), (2,6), (3,5), (4,4)\). Check whether every y-value is unique (i.e., no repeated y-values):\(8, 7, 6, 5, 4\) are all different.

Step 3: Confirm Function is One-to-One

Since all y-values are unique, the function is one-to-one. Thus, it can have an inverse.

Step 4: List Pairs of the Inverse Function

To find the inverse function, swap the x and y values in each pair of the original function. This results in: \((8,0), (7,1), (6,2), (5,3), (4,4)\).

Key Concepts

One-to-One Functions

A function is called a one-to-one function if each input maps to a unique output, and no two different inputs map to the same output. In practical terms, this means that every x-value (input) has one and only one y-value (output). Similarly, every y-value corresponds to one unique x-value. This unique mapping ensures that the function preserves its identity when inverted, meaning that if you switch the roles of inputs and outputs, the inverse remains a function.

To determine if a function is one-to-one, examine whether its y-values (outputs) are unique. In our case, the function pairs are

• (0,8)
• (1,7)
• (2,6)
• (3,5)
• (4,4)

If you analyze these pairs, notice that each y-value appears only once: 8, 7, 6, 5, and 4 are all distinct. This confirms that our given function is one-to-one. Therefore, it can have an inverse.

Horizontal Line Test

The Horizontal Line Test is a simple way to check if a function is one-to-one, to determine if it is invertible. Imagine drawing horizontal lines across the graph of a function. If any horizontal line touches the graph more than once, the function fails this test and cannot have an inverse.

This rule can be visualized for a set of function pairs as well. Consider if, for instance, multiple function pairs shared a y-value. For example, if both (2,5) and (3,5) were part of the function, a horizontal line at y=5 would intersect the function twice. This would indicate it's not one-to-one, and thus, does not have an inverse.

In our function, all y-values: 8, 7, 6, 5, and 4 are unique. Trying to draw a horizontal line at any of these y-values would touch only one pair at a time, ensuring that the function passes the Horizontal Line Test.

Function Pairs

Function pairs are the basic building blocks that show how a function works. In these pairs, the first number represents an x-value (input), and the second number is the y-value (output). These pairs not only describe the function but also help visualize the relationship between its sets of values.

For example, let's look at the pairs in the given function:

• (0,8)
• (1,7)
• (2,6)
• (3,5)
• (4,4)

Here, the inputs 0, 1, 2, 3, and 4 each map to a unique output, showing the one-to-one nature of the function. To find the inverse, simply swap each pair. This swapping reflects each input becoming an output and vice versa.

The inverse pairs are:

• (8,0)
• (7,1)
• (6,2)
• (5,3)
• (4,4)

This adjustment reveals the inverse function, maintaining the relationship between inputs and outputs consistently, yet with roles reversed.