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Chapter 1: Problem 20

Make an input-output table for the function. Use 1, 1.5, 3, 4.5, and 6 as the domain. $$ y=32-3 x $$

Short Answer

Expert verified
Input-output table: {(1, 29), (1.5, 27.5), (3, 23), (4.5, 18.5), (6, 14)}

Step by step solution

01

Substitution for x = 1

We substitute x = 1 into the function to find the corresponding y value: \(y = 32 - 3 * 1\)
02

Substitution for x = 1.5

Similar to the previous step, now substitute x = 1.5 into the function: \(y = 32 - 3 * 1.5\)
03

Substitution for x = 3

Substitute x = 3 into the function: \(y = 32 - 3 * 3\)
04

Substitution for x = 4.5

Substitute x = 4.5 into the function: \(y = 32 - 3 * 4.5\)
05

Substitution for x = 6

Finally, substitute x = 6 into the function: \(y = 32 - 3 * 6\)
06

Compile the Results

To finish, compile the pairs of x and y values that you computed in the previous steps. These will form your input-output table.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Input-Output Tables
Input-output tables are useful for visualizing how changes in input affect the output based on a specific function. In this example, the function given is a linear one: \(y = 32 - 3x\). An input-output table helps organize data by listing inputs (also known as the domain) alongside their corresponding outputs (the range).
Creating the table involves the following steps:
  • Choose a set of x-values, which will be the domain of the function. Here, the domain is {1, 1.5, 3, 4.5, 6}.
  • Substitute each x-value into the function to compute the corresponding y-value.
  • Fill in the table with x-values in one column and their computed y-values in another column.
This method offers a straightforward way to see the relationship between different values of x and their corresponding y, helping students grasp the concept of functions and how they work.
Domain and Range
Understanding the domain and range of a function is crucial when dealing with mathematical functions. For any function, the domain refers to the set of all possible input values, in this case, the x-values. The range is the set of all possible output values, or y-values, that result from substituting the domain values into the function.
For the given linear function \(y = 32 - 3x\), you can define:
  • The domain as the specific values provided: {1, 1.5, 3, 4.5, 6}.
  • The range consists of the y-values obtained by substituting these x-values into the function.
Identifying domain and range is essential because it frames the limits within which a function operates. This understanding helps in predicting and describing function behavior.
Substitution Method
The substitution method is a technique used to solve functions by replacing variables with known values to find unknowns. In this context, you substitute each x-value from the domain into the function to calculate the corresponding y-value.
Follow these steps to apply this method:
  • Take a value from the domain (e.g., x = 1, 1.5, etc.).
  • Insert the value into the given function (e.g., substituting x = 1 gives \(y = 32 - 3(1)\)).
  • Compute the result to find y.
  • Repeat for all domain values to create a complete input-output table.
This method is particularly helpful for linear functions because it systematically reveals how each input affects the output. By practicing substitution, students solidify their understanding of how linear equations operate in practical scenarios.

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Most popular questions from this chapter

EQUATIONS AND INEQUALITIES Match the verbal sentence with its mathematical representation. The sum of \(x\) and 16 is less than 32.

For which inequality is \(x=238\) a solution? (A) \(250 \geq x+12\) (B) \(250x+12\)

Match the problem with the formula needed to solve the problem. Then use the Guess, Check, and Revise strategy or another problem-solving strategy to solve the problem. Area of a rectangle \(\quad A=l w \quad\) Distance \(\quad d=r t\) Simple interest \(\quad I=P r t \quad\) Volume of a cube Temperature \(\quad C=\frac{5}{9}(F-32)\) Surface area of a cube \(S=6 s^{2}\) A piece of cheese cut in the shape of a cube has a volume of 27 cubic inches. What is the length of each edge of the piece of cheese?

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$\frac{c+5}{3} \leq 4 ; 3$$

Your school is building a new computer center. Four hundred square feet of the center will be available for computer stations. Each station requires 20 square feet. You want to find how many computer stations can be placed in the new center. You write the equation \(20 x=400\) to model the situation. What do \(20, x,\) and 400 represent? Solve the equation. Check your solution.
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