Problem 67 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The value of \((f \circ g) (2)\) is 11.

Step 1: Understand the Composition of Functions

The expression \((f \circ g)(2)\) means we first find \(g(2)\), and then use that result to evaluate \(f\). The composition of functions \((f \circ g)(x)\) is defined as \(f(g(x))\).

Step 2: Find the Value of g(2)

Look at the table for the row where \(x = 2\). There, \(g(2)\) is given as \(-3\).

Step 3: Evaluate f(g(2))

Now that we know \(g(2) = -3\), we need to find \(f(-3)\). Look at the table for the row where \(x = -3\). There, \(f(-3)\) is given as \(11\).

Step 4: Conclusion

Since \(f(-3) = 11\), the value of \((f \circ g)(2)\) is 11.

Key Concepts

Evaluating Functions

Evaluating functions involves determining the output of a function based on a given input, typically represented as \(x\). In simpler terms, we plug in a value for \(x\) and follow the function's rule to find the result.
When evaluating a function, follow these steps:

• Identify the function you are evaluating, such as \(f(x)\) or \(g(x)\).
• Determine the input value (e.g., 2 in the expression \(f(2)\)).
• Use the function's rule to find the corresponding output. This is often provided in a table or equation form.

To evaluate a function like \((f \circ g)(2)\), start by finding \(g(2)\) using the function table or rule. Then, take that result and evaluate \(f\) at that value. Here, \(g(2) = -3\) and so we look up \(f(-3)\), which is 11.
So, \((f \circ g)(2) = 11\). Understanding how to evaluate these compositions helps solidify comprehension of function operations.

Function Tables

Function tables are a useful tool for organizing and understanding functions. They display the relationship between inputs and outputs. In the table provided, each row lists an \(x\) value along with its corresponding \(f(x)\) and \(g(x)\) values.
For example, with the table given in the exercise:

• When \(x = -3\), \(f(x) = 11\) and \(g(x) = -8\).
• When \(x = 2\), \(f(x) = 1\) and \(g(x) = -3\).

Using a function table allows students to quickly find the output for any given input. This is particularly helpful in compositions of functions like \((f \circ g)(x)\), as it involves determining the result of one function and then applying it as the input for another.
It's essential to become comfortable navigating and interpreting these tables, as they are a staple in algebra. They simplify the process of evaluating functions and provide a visual way to understand how different functions may interact with each other.

Algebra Concepts

Algebra concepts form the foundation of understanding and working with functions. In algebra, functions are mappings from inputs to outputs, and evaluating them requires applying these foundational concepts effectively.
One of the key algebraic operations is function composition, denoted as \((f \circ g)(x)\). It signifies taking the output of one function (\(g(x)\)) and using it as the input for another (\(f(x)\)).

• This composition means upon finding \(g(x)\), its result is then input into \(f\).
• Mathematically, it is expressed as \(f(g(x))\).

In our example, we found that \(g(2) = -3\), which becomes the input for \(f\), leading to \(f(-3) = 11\). This demonstrates how function composition requires careful step-by-step evaluation, understanding both the order of operations and the function's characteristics.
Understanding these algebra concepts is crucial for navigating more advanced mathematical problems. They provide techniques for manipulating and interpreting mathematical information, which is foundational to many math-related fields and problems.