Problem 22 - Algebra 1

Question

Make an input-output table for the function. Use 1, 1.5, 3, 4.5, and 6 as the domain. $$ y=2+\frac{x}{0.5} $$

Step-by-Step Solution

Answer

The resulting input-output table is: (1,4), (1.5,5), (3,8), (4.5,11), (6,14)

Step 1: Understand the Function

The provided function is \(y=2+\frac{x}{0.5}\). In this function, x is the input and y is the output. Our task is to find the output (y) for various input (x) values.

Step 2: Substitute x=1 into the function

When we substitute x=1, we get \(y=2+\frac{1}{0.5}=2+2=4\)

Step 3: Substitute x=1.5 into the function

When we substitute x=1.5, we get \(y=2+\frac{1.5}{0.5}=2+3=5\)

Step 4: Substitute x=3 into the function

When we substitute x=3, we get \(y=2+\frac{3}{0.5}=2+6=8\)

Step 5: Substitute x=4.5 into the function

When we substitute x=4.5, we get \(y=2+\frac{4.5}{0.5}=2+9=11\)

Step 6: Substitute x=6 into the function

When we substitute x=6, we get \(y=2+\frac{6}{0.5}=2+12=14\)

Key Concepts

Input-Output Table

In mathematics, an input-output table helps display how a function relates each input value to an output value. This visual representation can make complex functions easier to understand. You start with a list of inputs, which are independent variables typically identified as "x". The corresponding outputs, often referred to as "y", are calculated using a given function.
For instance, in our function, \(y=2+\frac{x}{0.5}\), the input values are 1, 1.5, 3, 4.5, and 6. You substitute each input into the function to find the output:

• Input of 1 results in an output of 4
• Input of 1.5 results in an output of 5
• Input of 3 results in an output of 8
• Input of 4.5 results in an output of 11
• Input of 6 results in an output of 14

Creating a table out of this data aids in visualizing the relationship, simplifying the interpretation of the function's behavior across the specified domain.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Understanding these is crucial to evaluating functions. The given function, \(y=2+\frac{x}{0.5}\), is an algebraic expression where 'x' serves as the variable.
This expression implies that whatever value 'x' takes, the output is determined by performing the operations shown: first dividing \(x\) by 0.5, then adding 2 to the result.
Algebraic expressions can become more complex, but the core principle remains simple: apply arithmetic operations to variables and constants. Recognizing how each part of an expression influences the results helps in mastering various algebraic manipulations, enabling you to predict how changes to inputs affect outputs.

Substitution Method

The substitution method involves replacing a variable with a specific value to simplify the computation of a function. In our example, the function \(y=2+\frac{x}{0.5}\) is evaluated by substituting different values for 'x'. This process involves plugging each value of 'x' into the expression.
Let's break it down with examples:

• For \(x=1\), substitute 1 for 'x' to calculate \(y=2+\frac{1}{0.5}\), simplifying to \(4\).
• For \(x=1.5\), substitute 1.5 for 'x' to get \(y=2+\frac{1.5}{0.5}\), resulting in \(5\).
• Repeat this process for all given 'x' values to find their corresponding 'y' values.

Each step ensures accurate function evaluation, crucial for creating an effective input-output table and understanding the function's workings. This simple process gives unique results for different inputs, showcasing the function's dynamism.