Problem 78 - College Algebra with Modeling and Visualization

Question

Exercises 77 and 78: Numerical representations for the functions \(f\) and \(g\) are given. Evaluate the expression, if possible. $$ \begin{array}{llll} \text { (a) }(g \circ f)(1) & \text { (b) }(f \circ g)(4) & \text { (c) }(f \circ f)(3) \end{array} $$ $$ \begin{array}{rrrrr} x & 1 & 3 & 4 & 6 \\ f(x) & 2 & 6 & 5 & 7 \end{array} $$ $$ \begin{array}{rrrrr} x & 2 & 3 & 5 & 7 \\ g(x) & 4 & 2 & 6 & 0 \end{array} $$

Step-by-Step Solution

Answer

(g ∘ f)(1) = 4; (f ∘ g)(4) is undefined; (f ∘ f)(3) = 7.

Step 1: Understand Function Composition

Function composition \( (g \circ f)(x) \) means apply \( f(x) \) first, and then apply \( g \) to the result. Similarly, \( (f \circ g)(x) \) means apply \( g(x) \) first, and then apply \( f \) to the result. Evaluate each separately using the given tables.

Step 2: Evaluate (g ∘ f)(1)

First, find \( f(1) \) in the table for function \( f \), which is \( 2 \). Next, use this result as the input for function \( g \), so find \( g(2) \) in the table for function \( g \), which is \( 4 \). Hence, \( (g \circ f)(1) = 4 \).

Step 3: Evaluate (f ∘ g)(4)

Start with finding \( g(4) \). However, 4 is not listed as an input in the table for \( g \), meaning \( g(4) \) is undefined. Therefore, \( (f \circ g)(4) \) is also undefined.

Step 4: Evaluate (f ∘ f)(3)

First, find \( f(3) \) in the table for function \( f \), which is \( 6 \). Use this result as the new input, so find \( f(6) \), which is \( 7 \). Thus, \( (f \circ f)(3) = 7 \).

Key Concepts

Numerical Representations

When working with functions, it's important to understand how numerical representations help us extract valuable information about these functions. Think of numerical representations like a detailed map that guides you through the landscape of functions.

In this exercise, numerical tables are provided for the functions \( f(x) \) and \( g(x) \). These tables show the relationship between input values \( x \) and their corresponding outputs \( f(x) \) or \( g(x) \). For example, if \( x=1 \), the output for \( f(x) \) is \( 2 \), found as \( f(1)=2 \).

Having this organized in a table format simplifies the process of looking up values, which is practical especially when performing function evaluations or compositions. Each row represents a different set of inputs and outputs, helping us quickly follow the path from an input value to its resulting output.

Function Evaluation

Evaluating a function means finding the output value for a specific input, using the function's rule or table. It's like following a recipe where the input is your ingredient and the rule of the function is the recipe steps.

Let's consider the function \( f(x) \) from the exercise. To evaluate \( f(1) \), we look for the input \( x=1 \) in the table of function \( f \) and find its corresponding output, which is \( 2 \). This is denoted as \( f(1) = 2 \).

Function evaluation is straightforward when you have all the necessary inputs listed in the table. However, if an input value is missing, such as when trying to find \( g(4) \), the function is not defined for that value, and the evaluation is not possible.

Composite Functions

Composite functions involve combining two functions into one, by using the output of one function as the input of another. It's a bit like solving a puzzle where each piece (function) fits into the next.

Mathematically, a composite function \( (g \circ f)(x) \) means first calculating \( f(x) \), and then using this result as the input for \( g(x) \).

Using our exercise as an example, to evaluate \( (g \circ f)(1) \):

• First, locate \( f(1) \) in the table, which is \( 2 \).
• Next, take the result and find \( g(2) \) in the table, which is \( 4 \).
• Thus, \( (g \circ f)(1) = 4 \).

Composite functions require a careful step-by-step approach, ensuring each output becomes the new input in a chain. This process also demands that each stage in the sequence has a defined result to use at the next step.