Problem 45 - Precalculus : Building Concepts and Connections

Question

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \((g \circ f)(-1)\)

Step-by-Step Solution

Answer

The value for \(g \circ f(-1)\) is 0.

Step 1: Table Lookup for f(-1)

To find \(g \circ f(-1)\), first we need to find the value of \(f(-1)\). From the first table, we can see that \(f(-1) = -2\).

Step 2: Table Lookup for g(f(-1))

Now let's substitute that value into \(g\). We need to find \(g(f(-1)\), or \(g(-2)\). From the second table, we see that \(g(-2) = 0\).

Step 3: Solution for the Composite Function

Therefore, \(g \circ f(-1) = g(-2) = 0\).

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics where you combine two functions in such a way that the output of the first function becomes the input of the second. In notation, it is typically represented as \((g \circ f)(x)\), which is read as "g of f of x." Here, you'll evaluate the function \(f\) first and then use its output as the input for \(g\).
This process can be thought of as a kind of pipeline or chain reaction:

• You start with an initial value \(x\).
• Compute \(f(x)\) to get a result \(y\).
• Then use this \(y\) to find \(g(y)\).
• The result is \((g \circ f)(x)\).

This method allows you to create new functions that are derived from existing ones, offering a powerful way to solve complex problems by building on simpler ones.

Table of Values

A table of values is a straightforward way to represent the outcomes of a function. It shows specific input-output pairs, which allows you to look up the value of a function at particular points directly. This can simplify the process of solving function-related problems without having to compute function expressions each time.
For instance, in the exercise, we were given two tables, one for the function \(f\) and another for \(g\). Each row in these tables provides the value of the function for a specific input.

• For \(f(x)\), you simply find the row where \(x = -1\) and see that \(f(-1) = -2\).
• Next, use \(-2\) as the input for the \(g(x)\) table, leading to \(g(-2) = 0\).

Tables are especially useful for quickly evaluating compositions like \( (g \circ f)(x) \) because they provide direct access to the needed values.

Function Evaluation

Evaluating functions is about determining the output for a specific input. The function evaluation involves plugging the given input into the function and calculating the output according to the rule or expression that defines the function.
Using the tables provided in the exercise, function evaluation becomes a straightforward task. Here’s how it works:

• Look at the variable or the specific input you need to evaluate the function for, such as \(-1\) for \(f(x)\).
• Access the table to find the corresponding output, like \(f(-1) = -2\) in our case.
• Use this output to move to the next function, as we did by evaluating \(g\) at \(-2\).
• Continue this until all compositions or calculations are complete.

Understanding function evaluation helps in working through more complicated compositions and grasping how functions transform inputs into outputs.