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Chapter 2: Problem 76

Let \(f(x)=-2 x+5 .\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=-7\)

Short Answer

Expert verified
The value of \( x \) is 6.

Step by step solution

01

Understand the Problem

You are given a function \( f(x) = -2x + 5 \) and a target value of the function \( f(x) = -7 \). We need to find what value of \( x \) makes the output of the function equal to \( -7 \).
02

Set the Equation

Set the equation by substituting \( f(x) \) with \( -7 \). This gives us \( -2x + 5 = -7 \).
03

Isolate the Variable Term

Subtract \( 5 \) from both sides of the equation to move the constant term. This gives us \( -2x = -7 - 5 \), which simplifies to \( -2x = -12 \).
04

Solve for x

Divide both sides of the equation by \( -2 \) to solve for \( x \). This gives us \( x = \frac{-12}{-2} \).
05

Simplify the Solution

Simplify \( \frac{-12}{-2} \) to get \( x = 6 \). This is the value of \( x \) for which \( f(x) = -7 \).

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Key Concepts

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

Function Evaluation
Evaluating a function involves plugging a specific value into the function to see what output it produces. In the given exercise, we have the function \( f(x) = -2x + 5 \).
To evaluate this function at \( x \), you typically replace \( x \) with a specific number to find the output. However, in this problem, we are asked to do the reverse: we know the output \( f(x) = -7 \) and must find the value of \( x \) that results in this output.
  • Identify the function: \( f(x) = -2x + 5 \)
  • Given the output: \( f(x) = -7 \)
  • Task: Determine \( x \) such that the above function yields the given output.
This initial evaluation step helps set the stage for forming an equation that we can solve to find the unknown \( x \).
Solving Equations
When tasked with solving an equation like \( -2x + 5 = -7 \), our goal is to find the value of \( x \) that makes the equation true. Solving equations entails systematic steps to isolate the variable on one side.
First, we need to "set up" our equation by matching terms carefully: given \( f(x) = -7 \), we use the function definition to rewrite it as \( -2x + 5 = -7 \).
  • Starting equation: \( -2x + 5 = -7 \)
  • Our aim: simplify and manipulate this equation until it gives a definite solution for \( x \).
This structured approach ensures you proceed step-by-step, maintaining equality and accuracy throughout the solving process. Carefully carrying out each step is crucial because one small mistake can lead to an incorrect solution.
Variable Isolation
To isolate a variable means to rearrange an equation to get the variable by itself on one side. This is essential for solving equations.
In the exercise, after forming \( -2x + 5 = -7 \), the next step is to eliminate any constants that are not attached to \( x \).
  • Subtract \( 5 \) from both sides: This removes the constant on the side with the variable.
  • This simplifies the equation to \( -2x = -12 \).
  • Next, divide each side by \( -2 \) to solve for \( x \): \( x = \frac{-12}{-2} \).
  • Simplify the division to get \( x = 6 \).
By focusing on isolating \( x \), you ensure that your final result is accurate and confirms the correct value that makes the original function equal to \(-7\). This step-by-step breakdown is crucial in achieving clear solutions for linear equations.

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