Problem 61 - College Algebra

Question

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 3 to evaluate each expression. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline {x} & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline {f(x)} & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline {g(x)} & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0}\\\ \hline \end{array}$$ $$g(f(3))$$

Step-by-Step Solution

Answer

The value of \(g(f(3))\) is 4.

Step 1: Evaluate f(3)

According to the table, find the value of the function \(f\) when \(x = 3\). From the table, \(f(3) = 8\).

Step 2: Substitute f(3) into g

Now use the result from Step 1, which is \(f(3) = 8\), and substitute it into the function \(g\) to find \(g(f(3))\), which is \(g(8)\).

Step 3: Evaluate g(8)

Look up the value of \(g\) when \(x = 8\) using the table. The table shows \(g(8) = 4\).

Key Concepts

Composition of Functions

When we talk about the composition of functions, it might sound complex at first, but in reality, it's pretty straightforward! Function composition is essentially about plugging one function into another. In other words, you are using the output of one function as the input of another.

Imagine you have two functions, say \( f(x) \) and \( g(x) \). When you compose them, you might see it written as \( g(f(x)) \). This is read as "g of f of x". What this means is that you first calculate \( f(x) \), and then you put this result into \( g \) and calculate again.

• First, find \( f(x) \)
• Second, use the result in \( g(x) \)

This technique allows us to link two functions together and see their relationship dynamically. It’s like combining two machines where the output of one serves as the input for the other. Let’s explore how to do this step-by-step.

Step-by-Step Solution

Solving a composition of functions involves a simple and logical process. Let’s break it down step-by-step by using the given functions \( f(x) \) and \( g(x) \) from our example table.

**Step 1: Evaluate \( f(3) \)**
Find the value of the function \( f \) when \( x = 3 \). Look at the table and find \( f(3) \). As given, \( f(3) \) equals 8.

**Step 2: Substitute \( f(3) \) into \( g \)**
Now, take the result from the first step, which is \( f(3) = 8 \), and use it as the input for the function \( g \). This means you have to find \( g(8) \) now.

**Step 3: Evaluate \( g(8) \)**
Go back to the table and find the value of \( g \) when \( x = 8 \). Here, \( g(8) = 4 \).

• First, find \( f(3) = 8 \)
• Second, calculate \( g(f(3)) = g(8) \)
• Finally, get the result \( g(8) = 4 \)

It's like a relay race. First, you find \( f(x) \), then pass the baton of its result to \( g(x) \). Easy, isn't it?

Function Table Values

Using a function table is extremely useful when evaluating function expressions such as compositions! Whenever you have a set of predefined values for functions like \( f\) and \( g\), a table helps in quickly finding out results without having to calculate the function from scratch each time.

The table you'll use has two primary columns; the first column is the value of \( x \), and the second column contains the corresponding function values. The rows represent different inputs \( x \) and their results for specific functions.

• For \( f(x) \) values: simply align your \( x \) from the problem with the "\( f(x) \)" row values
• For \( g(x) \) values: do the same by aligning your result of \( f(x) \) to "\( g(x) \)" row

Not only do these tables save time, but they prevent mistakes by ensuring we're using accurate pre-calculated values. For instance, given \( f(x) \) and \( g(x) \) for inputs can help you focus on understanding function behaviors, enhancing your learning experience. This aids clarity, especially in function compositions, by making each query direct and efficient.