Problem 68 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value of \((g \circ f)(2)\) is 0.

Step 1: Understand the Composition

The notation \((g \circ f)(x)\) represents the composition of two functions. It means that you first apply the function \(f\) to \(x\), and then apply the function \(g\) to the result of \(f(x)\). In other words, \(g(f(x))\).

Step 2: Find f(2)

Look at the table to find the value of \(f(x)\) when \(x = 2\). According to the table, \(f(2) = 1\). This means that when you input \(2\) into the function \(f\), the output is \(1\).

Step 3: Find g(f(2))

Substitute the value of \(f(2)\) found in Step 2 into \(g\). Therefore, we need to find \(g(1)\).

Step 4: Find g(1)

Look at the table to find the value of \(g(x)\) when \(x = 1\). According to the table, \(g(1) = 0\). This means that when you input \(1\) into the function \(g\), the output is \(0\).

Key Concepts

Understanding Function Values

When we talk about function values, we're referring to the output you get when you plug a specific input into a function. Think of it like using a recipe. You have an input ingredient (the variable), you apply your recipe (the function), and then you get an output result. In mathematical terms, if you have a function like \( f(x) \), to find out what \( f(x) \) is for a given \( x \), you simply put that value in place of \( x \) in the function. In our exercise, function values are listed in a table to easily show what output (the function value) we get for each input. This makes finding and understanding function values straightforward.

The Process of Evaluating Functions

Evaluating a function means calculating its output for a specific input value. Let's break this down. You start with an input value, say \( x = 2 \), and insert it into the function to see what output it gives you. If you have a function \( f \) and an input \( x \), the "evaluation" process asks, "What is \( f(x) \)?"

Here’s a simple rundown of how you would evaluate functions based on our table:

• Identify your input value from the x-row.
• Find this input value in the function's table.
• The corresponding value is the function value or output.

With the above table, when evaluating \( f(2) \), we find the number directly from the column where \( x = 2 \), which gives us our function value of \( 1 \).

Using Tables for Functions

Tables for functions are an excellent tool for quickly finding function values without having to compute them manually every time. These tables list input/output pairs directly, making evaluation speedy and error-free. You'll typically see these as two-row tables: the first row for input values, \( x \), and the second for their corresponding outputs \( f(x) \) or \( g(x) \).

Here are some quick tips for using tables effectively:

• Always review the table header to understand which values belong where.
• Check both the row and column to find your desired output for a given input.
• Use the table to cross-check calculations performed mentally or manually.

In our example, the table neatly shows all values for \( f(x) \) and \( g(x) \) at different input values of \( x \), allowing us to perform function compositions like \( (g \circ f)(2) \) with ease.