Problem 69 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value is 0.

Step 1: Understanding the Composition of Functions

The expression \( (g \circ f)(3) \) represents the composition of the functions \( g \) and \( f \). First, we need to find \( f(3) \). Then, we substitute this result into \( g(x) \). This can be written as \( g(f(3)) \).

Step 2: Finding f(3)

We look at the table to find the value of \( f(x) \) when \( x = 3 \). According to the table, \( f(3) = -1 \).

Step 3: Substituting f(3) into g(x)

Now that we have \( f(3) = -1 \), we need to find \( g(f(3)) = g(-1) \). Using the table, we see that \( g(-1) = 0 \).

Step 4: Conclusion

Thus, the value of the expression \( (g \circ f)(3) \) is \( 0 \).

Key Concepts

Function Values

When working with functions, one of the primary tasks is finding their values at a specific input. Let's say we have a function \( f(x) \), which gives us an output for every input \( x \). The table for \( f \) shows us the function values that correspond to different inputs. For example, if we want to find \( f(3) \), we simply look at the row for \( x = 3 \) in the table and find that \( f(3) = -1 \).

This concept of function values is essential because it helps us understand what output we get for each input. It is like a rule or a map that tells us where each input goes in the function's world.

• Always check the table or the function's rule for the specific input value.
• Function values are unique for a given input in most well-defined functions.

Understanding function values is the first step in more complex ideas, like function composition.

Composition of Functions

Function composition involves combining two or more functions. It's like feeding the output of one function into the input of another. The notation \((g \circ f)(x)\) means we're first applying \( f(x) \), and then using that output as the input for \( g(x) \). It is crucial to perform these steps in the correct order.

Consider \((g \circ f)(3)\). First, determine \( f(3) \), and then use that result as the input for \( g \). If \( f(3) = -1 \), then you find \( g(-1) \). In our case, \( g(-1) = 0 \), so \((g \circ f)(3) = 0\).

• Composition often involves two simple steps: find the result of the first function, then pass it to the second.
• Function composition is not necessarily commutative, meaning \((g \circ f)(x) eq (f \circ g)(x)\) always.

Understanding how functions combine gives us powerful tools for dealing with more complex problems.

Evaluating Expressions

Evaluating expressions, especially involving functions, requires careful execution of mathematical operations. By evaluating, we mean substituting values into an expression to simplify it to a final answer. Let's take the composed function expression \((g \circ f)(x)\) as an example.

Start by determining the inner function's result, like finding \( f(3) \), which is \(-1\). Next, plug this result into the outer function: \( g(-1) \) equals \( 0 \). When evaluating, always ensure the order of operations is followed: first the inside, then the outside.

• Always find inner function values first before moving to the outer in compositions.
• Be consistent with the way the table values are utilized for accuracy.

By mastering the evaluation of expressions, you'll better navigate through complex problems with ease.