Problem 59 - College Algebra

Question

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

Step-by-Step Solution

Answer

f(g(5)) = 9

Step 1: Identify the inner function

First, evaluate the expression inside the function. Here, that expression is \(g(5)\).

Step 2: Find the value of \(g(5)\)

Using the table, locate the value in the row labeled \(g(x)\) that corresponds to \(x = 5\). From the table, \(g(5) = 8\).

Step 3: Substitute \(g(5)\) into the outer function

Now, substitute the value found for \(g(5)\) into the expression \(f(g(5))\), which becomes \(f(8)\).

Step 4: Find the value of \(f(8)\)

Using the table, locate the value in the row labeled \(f(x)\) that corresponds to \(x = 8\). From the table, \(f(8) = 9\).

Key Concepts

Function Tables

Function tables are a helpful tool for understanding the specific values that functions take on at given inputs. These tables allow you to see at a glance how a function behaves for various inputs.
In a function table, each column generally represents a specific input value, while the rows show the corresponding output for one or multiple functions.

• The top row usually contains the input values, which are often labeled as \(x\).
• The rows below contain the corresponding function outputs for each input value, such as \(f(x)\) or \(g(x)\).

By using function tables, you can quickly identify the output of a function without needing to compute it from scratch each time. This makes it easier to perform operations like evaluating and composing functions quickly.

Evaluating Functions

Evaluating functions involves finding the output value of a function given a specific input. This is a crucial step in understanding how a function operates.
When given a specific \(x\)-value, to evaluate a function:

• Find the input value on the table, often the topmost row.
• Look directly below this input in the function's row to find the corresponding output.

Suppose we want to evaluate \(g(5)\). By locating \(x = 5\) in the table, you will find your desired function output below, which is 8 in this case. So, \(g(5) = 8\).
Evaluating functions is a straightforward service of function tables and helps in making quick calculations and decisions.

Nested Functions

Nested functions, sometimes called composite functions, occur when one function is inside another. This is important for understanding more complex operations between functions.
In an expression like \(f(g(x))\), \(g(x)\) is the inner function calculated first, and its result is used as the input for the outer function \(f(x)\).
For example, to evaluate \(f(g(5))\):

• First, determine \(g(5)\) using the function table. We find \(g(5) = 8\).
• Next, use 8 as the input to \(f(x)\), leading to \(f(8)\).
• Evaluating \(f(8)\) via the table gives the result 9.

These steps illustrate how nested functions work seamlessly together, starting with the inner function and moving outward to reach the final result.