Problem 55 - College Algebra

Question

f and g are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{cc}x & f(x) \\ \hline-1 & 1 \\ 0 & 4 \\ 1 & 5 \\ 2 & -1 \end{array}$$ $$\begin{array}{cc}x & g(x) \\ \hline-1 & 0 \\ 1 & 1 \\ 4 & 2 \\ 10 & -1 \end{array}$$ $$ (g \circ f)(-1) $$

Step-by-Step Solution

Answer

The combined function \(g \circ f) at -1 is 1.

Step 1: Identify the value of f(-1)

Firstly, locate -1 in the x-column of the \(f(x)\) table to find the corresponding \(f(x)\) value. In the table, \(f(-1)\) equals 1.

Step 2: Compute \(g(f(-1))\)

With the value of \(f(-1)\) found in Step 1, apply this value to the function \(g(x)\), so we need to evaluate \(g(1)\). From the table for \(g(x)\), we can see that \(g(1)\) equals 1.

Step 3: Final Result.

This implies that the composite function \(g \circ f) at -1 is also 1.

Key Concepts

Function Evaluation

When you're dealing with function evaluation, you're basically plugging values into functions to see what you get out. Think of it like putting something into a machine and seeing what comes out the other end. For example, if you have a function \( f(x) \), you're interested in determining \( f(1) \), \( f(2) \), etc. You find these values by looking them up in a table or calculating them mathematically. In the context of the exercise, function evaluation involves identifying the output of the function \( f(x) \) for a specific input given by the x-values in the table provided. It's crucial to read the tables carefully, ensuring you match the correct input with its corresponding output. By pinpointing the value that \( f(x) \) assigns to a certain \( x \), you are employing function evaluation. This forms the foundation for more complex operations like composing functions.

Table of Values

A table of values is an incredibly useful tool for understanding functions, especially when you're dealing with multiple layers of calculations as in composite functions.

• Each row in the table shows a pair of \( x \) and \( f(x) \). This indicates what happens when you use the function on that \( x \) input.
• Tables help you keep track of each function's output, serving as a quick reference to avoid recalculating values unnecessarily.

In our exercise, we have two tables of values: one for \( f(x) \) and another for \( g(x) \). To find \( f(-1) \), we look in the \( f(x) \) table and find that it equals 1. The table makes it easy to spot this value without extra computation. The same process applies to the \( g(x) \) table when determining \( g(1) \) after evaluating \( f(-1) \). This combination of values from both tables is key to solving the composite function.

Function Composition

Function composition is a bit like creating a recipe by following a process—that is, using the output of one function as the input of another. If you have two functions, \( f(x) \) and \( g(x) \), composing them means you first use \( f(x) \) and then plug that result into \( g(x) \). We write this as \( (g \circ f)(x) \) and it signifies "\( g \) of \( f \) of \( x \)."

• Start by evaluating \( f(x) \) with your initial \( x \) value.
• The result of \( f(x) \) becomes the input for \( g(x) \).

For the problem at hand, we compute \( (g \circ f)(-1) \). This means we begin by finding \( f(-1) \) using the table, which gives us 1. Then we take this outcome and insert it into \( g(x) \), achieving \( g(1) \); from the table, it's clear that this equals 1. So, \( (g \circ f)(-1) = 1 \). Function composition is powerful because it combines two simple functions into a single, more function extension, letting you delve deeper into how different processes interact mathematically.