Problem 59 - College Algebra

Question

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 3 to evaluate each expression. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline {x} & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline {f(x)} & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline {g(x)} & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0}\\\ \hline \end{array}$$ $$f(g(5))$$

Step-by-Step Solution

Answer

9

Step 1: Understand the Expression

We are given the expression \( f(g(5)) \). This means we first need to find \( g(5) \) and then use this result to find \( f(g(5)) \).

Step 2: Find g(5)

From the table, locate the value corresponding to \( x = 5 \) in the row for \( g \). Here, \( g(5) = 8 \).

Step 3: Find f(g(5))

Now that we know \( g(5) = 8 \), we need to find \( f(8) \). Looking at the table, find the value of \( f \) when \( x = 8 \). Here, \( f(8) = 9 \).

Key Concepts

Function Notation

Function notation is a way to represent functions in mathematics using symbols. It provides a concise way to show relationships and operations involving functions. In function notation, you will typically see expressions like \( f(x) \) or \( g(x) \). Both of these notations mean a function named \( f \) or \( g \) with the variable \( x \) being the input.

The notation \( f(x) \) is used to indicate the output of the function \( f \) when the input is \( x \). Each unique input \( x \) gives a specific output \( f(x) \), forming a pair that is often represented using tables or graphs.

When dealing with composite functions, like \( f(g(x)) \), it's helpful to see this as two interconnected functions. First, process the innermost function \( g(x) \), then use the result as the input for the outer function \( f(x) \). This stepwise approach helps simplify evaluations and provides clarity on how compound expressions are broken down.

Table of Values

A table of values is commonly used to present the relationship between inputs (often \( x \)-values) and their corresponding outputs for a function. This format allows you to easily look up and understand how functions behave for specific inputs.

From the exercise, we see a table that lists inputs \( x \) from 0 to 9 and their corresponding values for two functions \( f(x) \) and \( g(x) \). Each row in the table shows how each function produces values for the same input \( x \). This makes it simple to evaluate and interpret functions using tabulated results.

• Column headers show input values \( x \).
• Rows for each function \( f(x) \) and \( g(x) \) show respective outputs.
• Using the table to find \( g(5) \) reveals \( g(5) = 8 \), and finding \( f(8) \) gives \( f(8) = 9 \).

Tables like this are useful for solving problems that involve function evaluations by providing a quick reference for input-output relationships.

Evaluating Functions

Evaluating functions involves finding the output for a given input. This requires using the function's definition, which might be a formula, graph, or table.

In our example, to evaluate \( f(g(5)) \), you must follow a process, often involving looking at intermediary functions within a composite function.

Here’s a step-by-step method:

• Identify the innermost function: Start with evaluating \( g(5) \).
• Locate \( g(5) \) in the table to find the output: Table shows \( g(5) = 8 \).
• Use the results as input for the next function: Consider \( f(8) \).
• Find \( f(8) \) in the table: Reveals \( f(8) = 9 \).

This method can be applied to any kind of functional expression, and it’s crucial because compound and composite functions rely on understanding the layering of operations to determine outcomes. By using this method, complex operations become manageable, and evaluations become clear.