Problem 71 - College Algebra

Question

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

Step-by-Step Solution

Answer

\((f \circ f)(3) = 7\).

Step 1: Understand the Notation

The notation \((f \circ f)(3)\) means we need to apply function \(f\) to \(f(3)\). This is a composition of the function \(f\) with itself.

Step 2: Find \(f(3)\)

Refer to the table provided. Look for the value in the row corresponding to \(x = 3\) under the column \(f(x)\). The value of \(f(3)\) is \(-1\).

Step 3: Find \(f(-1)\)

Using the result from Step 2, \(f(3) = -1\). Now, use \(x = -1\) to find \(f(-1)\) from the table. The value of \(f(-1)\) is \(7\).

Step 4: Compile the Solution

Thus, \((f \circ f)(3)\) is evaluated by finding \(f(f(3)) = f(-1)\). From Step 3, we found \(f(-1) = 7\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics where you determine the output of a function for a specific input value. Essentially, it involves substituting a given number for the variable in the function's expression to find the result. This substitution helps us understand how different inputs affect the output of a function, revealing patterns and behaviors.
Consider the function \( f(x) \). If you want to find \( f(3) \), you would look at the provided function table to determine what value corresponds to \( x = 3 \). In our situation, the table states that for \( x = 3 \), \( f(x) = -1 \). This tells us that when \( x \) is 3, the output of the function \( f \) is \(-1 \).

• Identify the function you are evaluating (e.g., \( f \) or \( g \)).
• Look for the specific input in the function table.
• Record the corresponding output value.

Appreciating function evaluation is key to understanding more complex functions and their compositions, as it forms the basis for exploring relationships between variables.

Function Table

A function table is a simple, organized way to display input-output relationships tied to functions. It lists values of \( x \) alongside the corresponding outputs of different functions, such as \( f(x) \) and \( g(x) \). This visual layout supports efficiency when working with multiple functions and inputs.
The beauty of a function table is that it provides immediate access to function outputs without solving equations manually. For instance, when tasked with finding \( f(3) \) or \( g(-2) \), a function table allows quick lookup of these values. This saves time and avoids calculation errors, making it easier to focus on solving complex expressions, like compositions.

• Ensures quick reference for outputs.
• Helps identify patterns or trends within the data.
• Makes function comparison straightforward by aligning values side by side.

Function tables are priceless tools in mathematics, welcomely simplifying tasks involving repeated evaluations or further analysis of relationships between inputs and outputs.

Composition Notation

Composition notation offers a way to describe combining two functions, where the output of one function becomes the input of another. The notation \((f \circ g)(x)\) describes this operation, meaning you apply function \(g\) to \(x\), and then function \(f\) to the result of \(g(x)\). This chaining together of functions facilitates more complex relationships modeling.
When dealing with composition notation, it's important to work in the proper order. For example, with \((f \circ f)(3)\), first, find \( f(3) \) using the function table. The resulting output is then used as the new input to the function \(f\) again. This is a concrete way to uncover how functions interact when they are combined.

• Start with the innermost function first (for nested compositions).
• Use function tables to find values for each step.
• Apply results sequentially to reach the final output.

Understanding composition notation is crucial to solving layers of functions, especially in calculus and advanced algebra, where functions intertwine to model sophisticated phenomena.