Problem 66 - College Algebra

Question

Use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$ \begin{array}{|c|r|r|r|r|r|r|r|} \hline \boldsymbol{x} & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 11 & 9 & 7 & 5 & 3 & 1 & -1 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & -8 & -3 & 0 & 1 & 0 & -3 & -8 \\ \hline \end{array} $$ $$ (f \circ g)(1) $$

Step-by-Step Solution

Answer

The value of \((f \circ g)(1)\) is 5.

Step 1: Understand the Composition of Functions

The problem requires evaluating the expression \((f \circ g)(1)\). This means we first need to find \(g(1)\) and then use that result as the input for the function \(f\).

Step 2: Evaluate g at x = 1

Consult the table to find the value of \(g(x)\) when \(x = 1\). The table gives \(g(1) = 0\).

Step 3: Evaluate f at g(1)

With \(g(1) = 0\), evaluate \(f\) at this value. From the table, \(f(0) = 5\).

Step 4: Substitute to Find (f ◦ g)(1)

Substitute the value \(f(g(1))\), which is \(f(0) = 5\). Therefore, \((f \circ g)(1) = 5\).

Key Concepts

Table of Values

To successfully evaluate functions like in this problem, a table of values is a handy tool. The table provided in the exercise lists corresponding values for two functions, \( f(x) \) and \( g(x) \), over a specified range of the variable \( x \). Each row shows what each function outputs when \( x \) takes different values. Breaking it down:

• Top row: different values of \( x \).
• Middle row: corresponding outputs for \( f(x) \).
• Bottom row: corresponding outputs for \( g(x) \).

By following this structure, you quickly find the necessary values needed for solving a composite function problem. For instance, you instantly see that \( g(1) = 0 \). Using tables is an efficient way to organize function values for clear and immediate reference.

Evaluate Functions

Once you've grasped how the table of values is set up, the next step is evaluating functions at specific points. In this exercise, we are particularly interested in evaluating composite functions, which can be a little tricky at first but become manageable once broken down into steps.

To evaluate \((f \circ g)(1)\), follow these two key steps:

• First, determine \( g(1) \). Simply refer to your table and extract the corresponding output for \( g \) when \( x = 1 \). From our table, we know \( g(1) = 0 \).
• Next, use the result from the previous step to find \( f(g(1)) \) or \( f(0) \). Again, check the table and find that \( f(0) = 5 \).

This approach simplifies the process by breaking down potentially complex calculations into clear, achievable steps. The key is understanding what each function tells you about \( x \) and then using these insights sequentially.

Step-by-Step Solution

A step-by-step solution is invaluable in guiding you through composite function problems. Let's walk through this exercise to clarify each action and the reasoning behind them:

1. **Understand the problem**: You are asked to evaluate \((f \circ g)(1)\). This signals a need to first deal with \( g(x) \).

2. **Evaluate \( g(x) \) at \( x = 1 \)**: By checking the table, you find \( g(1) = 0 \). This is crucial because it provides the input for the next function.

3. **Evaluate \( f(x) \) with \( g(1) \)**: Now, \( f(x) \) must be evaluated at this point. The table tells us \( f(0) = 5 \).

4. **Substitute and conclude**: Substitute \( f(g(1)) \) or substitute the values so that \( f(0) = 5 \) and therefore, \((f \circ g)(1) = 5 \).

By laying out each step clearly, it becomes easier to trace the logic and decisions during the process. This method prevents getting lost in the process and ensures clarity as each computation builds upon the last one.