Problem 55 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The result of \((g \circ f)(-1)\) is 1.

Step 1: Understand Compositions of Functions

A composition of functions (g ∘ f)(x) means we first apply f to x, and then apply g to the result.

Step 2: Apply the function f

We first need to find the value of \(f(-1)\) as per the data available in the table. The corresponding value of \(f(-1)\) from the table is 1.

Step 3: Apply the function g

Now, we need to find the value of \(g(f(-1))\) which actually translates to \(g(1)\). From the table, the value of \(g(1)\) is given as 1. This is our result.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, used when we need to determine the output of a function for a specific input value. Imagine you have a machine with a slot labeled 'Input' and a slot labeled 'Output.' When you place a particular number or value into the Input slot, the machine processes it and provides a result—this result is your function's evaluation.

In the exercise above, the functions \( f \) and \( g \) are represented in a way that uses tables to map input values to their corresponding outputs. For example, when evaluating \( f(x) \) for \( x = -1 \), we look up the value \( f(-1) \) in the table, finding the result 1. So in this specific evaluation, you know \( f(-1) = 1 \).

Function evaluation is also key to understanding function composition, where one function's output becomes another function's input.

Function Composition

Function composition is akin to linking two or more functions to create an entirely new function. One function feeds its output as the input to another function. Picture it as a multi-step process where the first step is applying one function, and the second step is applying the next function to the result of the first. This results in a compound effect or a synthesized function, often denoted as \( g \circ f(x) \).

The example above outlines a composition \( (g \circ f)(x) \), meaning you first find the output from \( f(x) \), then use that output as the input for \( g(x) \).

• First, evaluate \( f(-1) \) which yields 1.
• Then calculate \( g(1) \), which also yields 1.

This step-by-step combining of functions to form a composite function is a crucial technique in both pure mathematics and real-world applications, allowing complex processes to be managed in simpler sub-steps.

Table of Values

A table of values is a simple and effective way to represent a function. It lists input values paired with their corresponding output values. Using tables is especially useful in exercises like these because they provide a quick reference to determine function values for specific inputs.

In the given example, tables for \( f \) and \( g \) store pairings of \( x \) inputs with outputs \( f(x) \) and \( g(x) \) respectively. Here.

• The table indicates that \( f(-1) = 1 \)
• Separately, \( g(1) = 1 \)

By referring to the table of values, you can streamline the process of finding which output a function will produce from a given input. This way, any operations such as evaluations or compositions can be conducted more efficiently, helping you focus more on understanding the process rather than performing cumbersome calculations.