Problem 102 - Elementary and Intermediate Algebra

Question

Complete each table. See Example 11. $$ \begin{array}{|c|c|} \hline \text { Input } & {\text { Output }} \\ \hline x & {\frac{x}{3}+\frac{x}{4}} \\ \hline 12 & {} \\ \hline-36 & {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

12 leads to 7; -36 leads to -21.

Step 1: Understand the Function

The task requires finding the output values for given input values in the function expressed as \( f(x) = \frac{x}{3} + \frac{x}{4} \). This means for each input \( x \), the output is calculated by summing the fraction \( \frac{x}{3} \) and \( \frac{x}{4} \).

Step 2: Calculate Output for Input 12

Substitute \( x = 12 \) into the expression: \( \frac{12}{3} + \frac{12}{4} \). First, calculate \( \frac{12}{3} = 4 \) and \( \frac{12}{4} = 3 \). Sum these results: \( 4 + 3 = 7 \).

Step 3: Calculate Output for Input -36

Substitute \( x = -36 \) into the expression: \( \frac{-36}{3} + \frac{-36}{4} \). First, calculate \( \frac{-36}{3} = -12 \) and \( \frac{-36}{4} = -9 \). Sum these results: \( -12 + (-9) = -21 \).

Step 4: Record the Results in the Table

Using the calculations from the previous steps, complete the table with the output values for the given inputs: 12 and -36. For 12, the output is 7, and for -36, the output is -21.

Key Concepts

Function Evaluation

Function evaluation is the process of substituting a given value into a function to determine its output. Here, the function is expressed as \( f(x) = \frac{x}{3} + \frac{x}{4} \). To evaluate this function, simply replace \( x \) with the specific numerical input provided.

For example, if you have to evaluate the function for \( x = 12 \), plug in 12 where \( x \) appears, and calculate:

• \( \frac{12}{3} \) results in 4.
• \( \frac{12}{4} \) results in 3.
• Add these two values: 4 + 3 = 7.

This answer, 7, is the output for the input \( x = 12 \).
Function evaluation helps determine outputs for different inputs, giving a clear map of how inputs relate to outputs through the given function.

Fraction Operations

Fraction operations involve a variety of actions such as addition, subtraction, multiplication, or division of fractions. In the context of our example, we focus on the addition of fractions:

When adding fractions like \( \frac{x}{3} \) and \( \frac{x}{4} \), we must find a common denominator.

Since 3 and 4 are the denominators in the expression \( f(x) = \frac{x}{3} + \frac{x}{4} \), their least common denominator is 12. Therefore, rewrite each fraction with the common denominator of 12:

• For \( \frac{x}{3} \), multiply the numerator and denominator by 4 to get \( \frac{4x}{12} \).
• For \( \frac{x}{4} \), multiply the numerator and denominator by 3 to get \( \frac{3x}{12} \).

Add these fractions: \( \frac{4x}{12} + \frac{3x}{12} = \frac{7x}{12} \). This simplifies the initial step when calculating specific outputs, ensuring easy calculation of function results.

Input-Output Tables

Input-output tables are helpful tools in mapping how different inputs into a function produce corresponding outputs. They serve as a clear and organized method to display the relationship between inputs and function results.

In the exercise given, the input-output table lists specific values for \( x \) and requires you to compute their respective outputs using the function \( f(x) = \frac{x}{3} + \frac{x}{4} \).

For example, with an input of 12, by substituting 12 in the function as shown:

• Calculate the output as 7, then record "12" under input and "7" under output.

Similarly, substitute -36 into the function and calculate the output as -21. Record these results accordingly.
These tables not only simplify the calculation process but also make it easier to see how the function processes various inputs to generate corresponding outputs.