Problem 69 - College Algebra

Question

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

Step-by-Step Solution

Answer

\((g \circ f)(3) = 0\).

Step 1: Understand Composition of Functions

The composition of functions \((g \circ f)(x)\) means that you first apply the function \(f\) to \(x\), and then apply the function \(g\) to the result from \(f\). In this exercise, we need to evaluate \((g \circ f)(3)\).

Step 2: Evaluate f(3)

Look at the table to find the value of \(f(3)\). According to the table, \(f(3) = -1\).

Step 3: Evaluate g(f(3))

Now that we know \(f(3) = -1\), we need to evaluate \(g(-1)\). According to the table, \(g(-1) = 0\).

Step 4: Conclude (g ∘ f)(3)

Based on our calculations, \((g \circ f)(3) = g(f(3)) = g(-1) = 0\).

Key Concepts

Table of Values

When dealing with functions, a table of values is an immensely helpful tool. It helps us quickly find the outcome of a function for a specific input without solving a function analytically. This table offers a visual and numerical reference at a glance.

In the context of function composition, tables allow students to understand how two functions interrelate by ensuring you can track function values easily for combination tasks. For browsing this table, ensure you:

• Identify the correct row and column intersection for your input values.
• Use these identified values as intermediate results for further calculations or compositions.

In the original exercise, values for two functions, \( f(x) \) and \( g(x) \), are provided against different \( x \) values. This structured presentation allows seamless retrieval of values like \( f(3) = -1 \) and \( g(-1) = 0 \).

In conclusion, tables ease the computational load and serve as a reference guide, especially when working with multiple expressions or complex compositions.

Evaluate Expressions

Evaluating expressions involves determining the value of an expression for given input values. This process is critical in mathematics as it involves substitution and applying known rules or relationships.

For our exercise, we need to evaluate the expression \((g \circ f)(3)\):

• First, substitute the inner function, \( f \), with the given input \( x = 3 \). From the table, we see \( f(3) = -1 \).
• Then substitute \( -1 \) into the outer function, \( g \). As per the table, \( g(-1) = 0 \).

It’s important to follow the specified order of functions when evaluating compositions to avoid mistakes. By systematically executing evaluations with the aid of a table, students can simplify expressions accurately and confidently.

In summary, evaluating expressions in this framework is an exercise in accuracy and order, ensuring each step respects the sequence of function operations.

Step-by-step Solution

Providing solutions in a step-by-step manner is crucial for learning, especially with function compositions, to ensure concepts are understood wholly and correctly.

Here is a comprehensive breakdown of the solution for the expression \((g \circ f)(3)\):

• Step 1: Understand Composition of Functions – Recognize that \((g \circ f)(x)\) means to first apply \( f(x) \) then apply \( g \) to the result.
• Step 2: Evaluate \( f(3) \) – Consult the table for \( f(x) \) to find that \( f(3) = -1 \).
• Step 3: Evaluate \( g(f(3)) \) – Use \( g(-1) \) from \( g(x) \)'s table, finding \( g(-1) = 0 \).
• Step 4: Conclude \((g \circ f)(3)\) – Compile findings: \( g(f(3)) = 0 \).

This logical sequencing not only helps in reaching the correct answer but enhances problem solving skills by highlighting function relationships and dependencies.

Overall, a step-by-step solution demystifies the evaluation process and empowers students to tackle more complex function compositions effectively.