Problem 51 - A Graphical Approach to College Algebra

Question

Tables for functions \(f\) and \(g\) are given. Evaluate each expression, if possible. (a) \((g \circ f)(1)\) (b) \((f \circ g)(4)\) (c) \((f \circ f)(3)\) $$\begin{array}{c|cc|c}\boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) & \boldsymbol{x} & \boldsymbol{g}(\boldsymbol{x}) \\\\\hline 1 & 4 & 1 & 2 \\\\\hline 2 & 3 & 2 & 3 \\\\\hline 3 & 1 & 3 & 4 \\\\\hline 4 & 2 & 4 & 5 \end{array}$$

Step-by-Step Solution

Answer

(a) 5, (b) undefined, (c) 4

Step 1: Evaluate \( (g \circ f)(1) \)

To find \( (g \circ f)(1) \), first evaluate \( f(1) \). From the table of \( f \), \( f(1) = 4 \). Next, use this value as the input for \( g \): evaluate \( g(4) \). From the table of \( g \), \( g(4) = 5 \). Thus, \( (g \circ f)(1) = 5 \).

Step 2: Evaluate \( (f \circ g)(4) \)

To find \( (f \circ g)(4) \), first evaluate \( g(4) \). From the table of \( g \), \( g(4) = 5 \). Now, we need \( f(5) \). However, there is no \( f(5) \) available in the given table, making the expression \( (f \circ g)(4) \) undefined.

Step 3: Evaluate \( (f \circ f)(3) \)

To find \( (f \circ f)(3) \), first evaluate \( f(3) \). From the table of \( f \), \( f(3) = 1 \). Now use this result to evaluate \( f(1) \). From the table of \( f \), \( f(1) = 4 \). Thus, \( (f \circ f)(3) = 4 \).

Key Concepts

Function Evaluation

Function evaluation is a basic yet crucial concept in mathematics. It involves finding the output of a function for a given input, using either a formula or a table of values. When you evaluate a function, you are essentially asking, "What result does this function produce when I input this particular value?"

For example, you might have a function \( f(x) \), and you want to evaluate \( f(2) \). You look at the function's definition and determine what outcome corresponds to the input of 2.

• If you have a table, simply find the row where \( x = 2 \) and read off \( f(x) \).
• If the function is given by an equation, substitute \( x = 2 \) into that equation and solve for \( f(x) \).

Function evaluation is a foundational skill that is often used as a stepping stone for understanding more complex concepts, such as composite functions.

Input-Output Table

An input-output table is a visual representation used to organize function data. It lists pairs of inputs and their corresponding outputs. This setup allows you to quickly determine what a function does with certain inputs.

Such tables make evaluating functions particularly straightforward. Here's how they work:

• The left column (or side) typically lists the input values (commonly labeled \( x \)).
• The right column (or side) lists the outputs, which are the results of applying the function to each input (e.g., \( f(x) \) or \( g(x) \)).

By reading across the table, you can directly identify the output for any given input—useful for calculations and understanding function behavior.

For complex problems like composite functions, tables are indispensable for tracing the calculations step-by-step.

Composite Function

Composite functions involve applying one function to the results of another. This concept is represented by the notation \((g \circ f)(x)\), which means you first apply \( f \) to \( x \), and then apply \( g \) to the result of \( f(x) \).

Creating composite functions involves a series of evaluations:

• Find \( f(x) \) using the input value \( x \).
• Use the output of \( f(x) \) as the new input for \( g \).
• Find \( g(f(x)) \) to get your final result.

One challenge with composite functions is ensuring the second function \( g \) is defined for the output of \( f(x) \). If not, the composite function is undefined at that point.

This process underlines the interconnected nature of these mathematical operations and showcases how functions can be chained together to solve complex equations.