Problem 1 - Precalculus: Functions and Graphs

Question

Find \(\begin{array}{ll}\text { (a) } f^{-1}(5) & \text { (b) } g^{-1}(6)\end{array}\) $$\begin{array}{|c|c|c|c|} \hline x & 2 & 4 & 6 \\ \hline f(x) & 3 & 5 & 9 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|} \hline x & 1 & 3 & 5 \\ \hline g(x) & 6 & 2 & 6 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

f^{-1}(5) = 4; g^{-1}(6) = 1 \, \text{or} \, 5

Step 1: Understand Inverses

To find the inverse of a function value, we need to determine which original input (x) resulted in the given output. For function values in the table, 'f(x)' and 'g(x)' give us outputs for various x inputs.

Step 2: Find f^{-1}(5)

We want to find which x value in the table corresponds to f(x) = 5. Looking at the table: - When x = 2, f(x) = 3. - When x = 4, f(x) = 5. - When x = 6, f(x) = 9. Thus, f^{-1}(5) = 4 because when x = 4, f(x) is 5.

Step 3: Identify the output for g^{-1}(6)

To find g^{-1}(6), we look for which x value corresponds to g(x) = 6 in the table: - When x = 1, g(x) = 6. - When x = 3, g(x) = 2. - When x = 5, g(x) = 6. In this case, g(x) = 6 at x = 1 and x = 5, so g^{-1}(6) could be either 1 or 5. The problem does not specify conditions for multiple results, so we include both.

Key Concepts

Function Tables

Function tables are a convenient way to organize and display the relationship between two variables, typically denoted as the input (x) and the output (f(x) or g(x)). In the tables provided for this exercise, each row represents a different function, and each column corresponds to a specific input value of x and the resulting output value of f(x) or g(x).

To effectively use a function table, identify the two main components:

• The input column, which lists the x values.
• The output column, which shows the corresponding function values like f(x) or g(x).

Looking at these values allows us to understand how changes in the input affect the output, which is essential when finding inverse functions. In this way, function tables simplify the process of identifying which input corresponds to a given output, thus facilitating the determination of inverse function values.

Function Values

Function values are the outputs that arise from substituting specific inputs into a given function. These values are vital for understanding the behavior of a function across its domain. To determine a function value, you substitute an x-value into a function and calculate the result. For example, in the function table for f(x), if x = 4, the output is f(x) = 5.

Here's how it works practically:

• Locate the desired x-value in the function table.
• Find the corresponding f(x) or g(x) value in the same row.
• Consider this f(x) or g(x) value as the function's output for that particular input.

Function values help in calculating inverse functions, as the inverse function essentially swaps the roles of x and the function output, allowing us to trace the original input that led to a specific output in the table.

Input-Output Relationships

Input-output relationships describe how a change in the input of a function leads to a change in the output. This relationship is the essence of how functions operate. Each input value has a corresponding output, and understanding this match is critical when dealing with inverse functions.

For example:

• In the f(x) function table, the input x = 4 results in an output of f(x) = 5, establishing a clear input-output relationship: when x is 4, the function gives 5.
• In g(x), the inputs x = 1 and x = 5 both result in an output of g(x) = 6, indicating that the output 6 is related to both inputs 1 and 5.

This relationship becomes particularly interesting when reversing the direction and looking for inverses. In inverse functions, we're essentially asking: given a certain output, what was the original input? Such queries hinge on the clarity of the input-output relationships captured in the function tables.