Problem 67 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result is 11.

Step 1: Identify the Inner Function

The expression \((f \circ g)(2)\) involves a composition of functions, meaning you first need to evaluate \(g(2)\). Locate \(x = 2\) in the row for \(g(x)\) in Table 4.

Step 2: Evaluate the Inner Function

In the table, \(g(2) = -3\). Thus, the result of the inner function evaluation is \(g(2) = -3\).

Step 3: Identify the Outer Function

Now that we have \(g(2) = -3\), substitute this result into the outer function to evaluate \(f(g(2))\), which is \(f(-3)\).

Step 4: Evaluate the Outer Function

Locate \(x = -3\) in the row for \(f(x)\) in Table 4. There you find that \(f(-3) = 11\).

Step 5: Conclude the Composition Evaluation

Hence, after evaluating both the inner and the outer functions, we find that \((f \circ g)(2) = 11\).

Key Concepts

Evaluating Functions

Evaluating functions is a mathematical process where you determine the output of a function given a specific input. In this exercise, the focus is on evaluating function compositions, which involves two functions, say, \(f\) and \(g\). For the expression \((f \circ g)(2)\), we apply the function \(g\) first and then use the result to evaluate \(f\). This implies you need to find the output of \(g\) when the input is 2, followed by determining what \(f\) outputs when given the result from \(g\).

Let's take a step-by-step approach for clarity. Start by identifying your inner function, which in this case is \(g(2)\). You chase that through the table of values provided, find \(g(2)\), and note its value. \(g\) gives you an answer that becomes the new input to your outer function, which here is \(f\). Again, you check the table, replace \(x\) with this new number in \(f(x)\), and retrieve that result. This involves a simple substitution process, but it requires careful attention to the function and whether the substitution affects \(f\) or \(g\).

Table of Functions

The table of functions is a vital tool in evaluating expressions as it provides specific values of functions at given inputs. Typically, each row or column represents inputs, and the corresponding row or column shows the function's output at that input. This allows you to quickly check the value that a function like \(f(x)\) or \(g(x)\) takes at a specified \(x\).

With reference to our exercise, we utilize the table to understand the values of \(f\) and \(g\) over a range of \(x\) values. For instance, when you wish to find \(g(2)\), you look down \(x=2\) under \(g(x)\), which gives you the output of \(-3\). In a similar manner, after obtaining \(-3\) as \(g(2)\), you move to locate \(x=-3\) in \(f(x)\) to compute \(f(g(2))\) which in this exercise equates to \(f(-3) = 11\).

• The table makes it easier to avoid arithmetic errors.
• It organizes your inputs and the corresponding outputs neatly.
• It acts as a quick reference for verifying recalculations.

Step by Step Solutions

To solve function composition slices like \((f \circ g)(2)\), a step-by-step approach ensures thorough understanding and correct answers. Here are the steps broken down:

• Step 1: Identify the inner function. Begin by looking for the function you're dealing with first, in this case, \(g(2)\).
• Step 2: Evaluate the inner function. Use the table to find out that \(g(2) = -3\).
• Step 3: Substitute this result into the outer function. Here, \(-3\) becomes the input to \(f\).
• Step 4: Evaluate the outer function. Locate \(x = -3\) in the \(f(x)\) column to find out that \(f(-3) = 11\).
• Step 5: Conclude the evaluation. Thus, \((f \circ g)(2)\) results in \(11\).

Each of these steps involves finding values in the table and understanding the order of operations—important building blocks in math competency. By proceeding one step at a time, you ensure each part of the problem is captured accurately and efficiently.