Problem 93 - College Algebra with Modeling and Visualization

Question

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ g^{-1}(4) $$

Step-by-Step Solution

Answer

\( g^{-1}(4) = 3 \)

Step 1: Understand Inverse Functions

The inverse of a function, denoted as \( g^{-1}(x) \), reverses the input and output of the original function \( g(x) \). To find \( g^{-1}(4) \), we need to find the value of \( x \) such that \( g(x) = 4 \).

Step 2: Locate the Output Value 4 in g(x)

Look at the second table where the function \( g(x) \) is defined. We need to find where \( g(x) = 4 \). This is solved by locating the value 4 in the second row labeled \( g(x) \).

Step 3: Identify the Corresponding x Value

Upon inspecting the \( g(x) \) table, we find that \( g(x) = 4 \) when \( x = 3 \). This implies the inverse function evaluated at 4, \( g^{-1}(4) \), gives us \( x = 3 \).

Step 4: Conclusion

We conclude that \( g^{-1}(4) = 3 \), meaning when the output of \( g(x) \) is 4, the input was 3.

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. Think of it as working through a math problem where you replace the variable with a number to compute the result.

When you have a function, like \( f(x) \) from the exercise, it means that for a specific \( x \) value, there is a corresponding \( f(x) \) value. For instance, if \( x = 2 \), as seen in the first table, \( f(x) = 5 \). This tells us that when we "plug in" 2 into the function \( f \), the result, or output, is 5.

Understanding function evaluation is crucial because it helps in determining what values you get from a function for any input you might choose. It acts like a guide showing the relationship between inputs and outputs in a function.

Table of Values

A table of values is a tool that makes it easier to understand how a function behaves by showing specific input-output pairs. It precisely lists what output each input produces, functioning like a map between \( x \) values and their corresponding \( f(x) \) or \( g(x) \) values.

For example, in the exercise above, there are two tables, one for \( f(x) \) and another for \( g(x) \). Each row corresponds to the inputs (\( x \) values) and outputs (\( f(x) \) or \( g(x) \) values) of the functions.

Tables are particularly useful for visually locating specific values, such as finding what input makes \( g(x) = 4 \). In this exercise, we use the table to determine that when \( x = 3 \), \( g(x) \) is 4, which is essential for understanding the inverse as we've found the output that matches our requirements.

Inverse Function Notation

Inverse functions are valuable because they help us "work backwards." If a function \( g(x) \) tells us what output \( y \) results from an input \( x \), the inverse function, \( g^{-1}(x) \), does the opposite. It identifies the starting input \( x \) when given the output \( y \).

The notation \( g^{-1}(x) \) indicates an inverse function, sometimes leading to confusion. It's not an exponent but a symbol to show the function that undoes \( g(x) \). For practice, in the exercise provided, we found \( g^{-1}(4) \). Using the \( g(x) \) table, we determine that when the result (\( g(x) \)) is 4, the input \( x \) must have been 3.

Learning to read and understand inverse functions is crucial. It equips you with the ability to transition back to initial values from produced outcomes, enhancing the comprehension of functional relations.