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Chapter 2: Problem 69

If \(f(x)=0.1 x-0.5,\) for what input is the output \(-3 ?\)

Short Answer

Expert verified
The input is -25.

Step by step solution

01

Identify the Given Equation and Output

The function provided is given by: \[ f(x) = 0.1x - 0.5 \]We are asked to find the input value (x) when the output is -3.
02

Set Up the Equation

Set the equation equal to the given output value: \[ 0.1x - 0.5 = -3 \]
03

Isolate the Term with x

Add 0.5 to both sides of the equation to begin isolating the x term: \[ 0.1x - 0.5 + 0.5 = -3 + 0.5 \]This simplifies to: \[ 0.1x = -2.5 \]
04

Solve for x

Divide both sides of the equation by 0.1 to solve for x: \[ x = \frac{-2.5}{0.1} \]Simplifying this gives: \[ x = -25 \]
05

Verify the Solution

Substitute x = -25 back into the original function: \[ f(-25) = 0.1(-25) - 0.5 = -2.5 - 0.5 = -3 \]The output is -3, which verifies that our solution is correct.

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Key Concepts

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

Function Notation
Function notation is a way to signify a function in mathematics. Instead of writing equations in a typical format like y = mx + c, function notation uses symbols.
For example, an equation like y = 0.1x - 0.5 can be written in function notation as f(x) = 0.1x - 0.5.
This means the function f depends on the variable x.
Using function notation makes it easier to represent complex relationships and solve equations.
Linear Functions
A linear function is a type of function where the relationship between the input (x) and the output (f(x) or y) is a straight line when graphed.
It has the general form f(x) = mx + b.
The 'm' stands for the slope, which indicates how steep the line is, and the 'b' stands for the y-intercept, where the line crosses the y-axis.
In our exercise, 0.1 is the slope (m) and -0.5 is the y-intercept (b).
Linear functions are simple yet powerful tools for modeling relationships in algebra.
Solving for x
Solving for x means finding the value of x that makes the equation true.
When given an equation like 0.1x - 0.5 = -3, you need to manipulate it to isolate x.
First, you would add 0.5 to both sides: 0.1x - 0.5 + 0.5 = -3 + 0.5, simplifying to 0.1x = -2.5.
Next, divide both sides by 0.1: x = -2.5 / 0.1, which gives x = -25.
The value -25 is the solution that balances the original equation.
Verifying Solutions
Verifying a solution means checking if your calculated value is correct.
To verify x = -25 in our function f(x) = 0.1x - 0.5, substitute -25 back into the function.
You get f(-25) = 0.1(-25) - 0.5 = -2.5 - 0.5 = -3.
The output matches the provided output of -3, so the solution is verified.
Always verify solutions to ensure accuracy and understanding.
This step confirms that your steps and calculations were correct.

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