Problem 35 - College Algebra with Modeling and Visualization

Question

Exercises 35 and 36: Use the tables to evaluate each expression, if possible. (a) \((f+g)(-1)\) (b) \((g-f)(0)\) (c)\((g f)(2)\) (d)\((f / g)(2)\) $$ \begin{array}{rrrr} x & -1 & 0 & 2 \\ f(x) & -3 & 5 & 1 \end{array} $$ $$ \begin{array}{rrrr} x & -1 & 0 & 2 \\ g(x) & -2 & 3 & 0 \end{array} $$

Step-by-Step Solution

Answer

(a) -5; (b) -2; (c) 0; (d) undefined.

Step 1: Evaluate Expression (f+g)(-1)

To find \(f+g\)(-1), add the values of \(f(x)\) and \(g(x)\) at \(x = -1\). According to the tables: \( f(-1) = -3\) and \(g(-1) = -2\). Now calculate: \(f(-1) + g(-1) = -3 + (-2) = -5\).

Step 2: Evaluate Expression (g-f)(0)

To find \(g-f\)(0), subtract the value of \(f(x)\) from \(g(x)\) at \(x = 0\). According to the tables: \(f(0) = 5\) and \(g(0) = 3\). Now calculate: \(g(0) - f(0) = 3 - 5 = -2\).

Step 3: Evaluate Expression (gf)(2)

To find \(g f\)(2), multiply the values of \(f(x)\) and \(g(x)\) at \(x = 2\). According to the tables: \(f(2) = 1\) and \(g(2) = 0\). Now calculate: \(f(2) \times g(2) = 1 \times 0 = 0\).

Step 4: Evaluate Expression (f/g)(2)

To find \(f/g\)(2), divide the value of \(f(x)\) by \(g(x)\) at \(x = 2\). According to the tables: \(f(2) = 1\) and \(g(2) = 0\). Since dividing by zero is undefined, \(f/g\)(2) is undefined.

Key Concepts

Function Evaluation

Function evaluation involves determining the output of a function for a given input. Think of functions as machines where you input a number, and the function "processes" this number and outputs a result. Using tables makes it easier to evaluate functions at specific points.
For example, to evaluate \( f(-1) \), you simply find the row in the table where \( x = -1 \) and pick the corresponding value of \( f(x) \). In this exercise, \( f(-1) = -3 \). Similarly, for \( g(x) \) using \( x = 0 \), \( g(0) = 3 \).
Function evaluation is the foundation of understanding how functions work, as it builds the confidence needed to perform operations using these values directly from their tables.

Addition of Functions

When adding functions such as \( (f+g)(x) \), you are essentially combining their outputs for the same input value. This is written as \( (f+g)(x) = f(x) + g(x) \). It's like adding individual results for a shared input to get a total result.
For instance, to evaluate \( (f+g)(-1) \), look up \( f(-1) \) and \( g(-1) \) from the tables: \( f(-1) = -3 \) and \( g(-1) = -2 \). The computation will be \( -3 + (-2) = -5 \).
This concept helps analyze how changes in one function influence the collective outcome when functions are combined.

Subtraction of Functions

Subtraction of functions, such as \( (g-f)(x) \), involves finding the difference in their individual outputs at the same input value. This is expressed as \( (g-f)(x) = g(x) - f(x) \).
Let's consider \( (g-f)(0) \): From the tables, \( g(0) = 3 \) and \( f(0) = 5 \). The calculation would be \( 3 - 5 = -2 \).
Grasping subtraction of functions offers insights into how the changes in one function can create relative differences with another, giving a more dynamic view of function interdependencies.

Multiplication of Functions

Multiplication of functions requires multiplying their outputs at a given input value. Represented as \( (gf)(x) = f(x) \times g(x) \), this operation explores the product of two function outputs.
For \( (gf)(2) \): From the function tables, \( f(2) = 1 \) and \( g(2) = 0 \), so \( 1 \times 0 = 0 \).
Multiplication of functions reveals how the effect of one function can amplify or nullify the result of another function. It's a crucial tool in analyzing the combined impact of multiple factors in mathematical modeling.

Division of Functions

Division of functions involves dividing the output of one function by another at a particular input. This operation, expressed as \( (f/g)(x) = \frac{f(x)}{g(x)} \), is slightly more complex due to the possibility of division by zero.
For instance, \( (f/g)(2) \) looks for \( f(2) = 1 \) and \( g(2) = 0 \). However, division by zero is not defined in mathematics, rendering \( (f/g)(2) \) undefined.
It's vital to acknowledge division of functions can uncover fundamental limitations within mathematical models and may require further methods to resolve these undefined expressions when they occur.