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Chapter 8: Problem 75

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

Short Answer

Expert verified
The value of \(x\) is 0.

Step by step solution

01

Understanding the Equation

The problem gives us a function \(f(x) = -2x + 5\) and tells us that the function has a value of 5. This means that we need to solve the equation \(f(x) = 5\) to find the value of \(x\). In mathematical terms, this is written as \(-2x + 5 = 5\).
02

Setting Up the Equation

First, write the equation with the given function value plugged in: \(-2x + 5 = 5\). This equation needs to be solved to find \(x\).
03

Isolating the Variable Term

Subtract 5 from both sides of the equation to help isolate the term containing \(x\): \(-2x + 5 - 5 = 5 - 5\). This simplifies to \(-2x = 0\).
04

Solving for \(x\)

Divide each side of the equation by \(-2\) to solve for \(x\). So, \(x = \frac{0}{-2}\). This simplifies to \(x = 0\).
05

Verification

Substitute \(x = 0\) back into the original function to check: \(f(0) = -2(0) + 5 = 5\). Since this equals the given function value, our solution is verified.

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Key Concepts

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

Function Evaluation
In the context of algebra, evaluating a function is a crucial technique that helps us determine the output of a function for a specific input. A function, like the one given in the problem, is generally written as:
  • Function notation: An example is \(f(x) = -2x + 5\)
    Here, \(f\) represents the function, and \(x\) is the input variable.
To evaluate a function, you plug in the value given for \(x\) into the equation and compute the result. In simpler terms, you replace \(x\) in the function with the given value to see what number or expression \(f(x)\) will become. This process helps you understand how the function behaves with different inputs and is essential in solving algebraic problems.
Solving Linear Equations
Linear equations are equations that make a straight line when graphed and are usually expressed as \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. To solve a linear equation for \(x\), the goal is to isolate \(x\) on one side of the equation. This involves a few key steps:
  • Step 1: Simplify each side of the equation if necessary.
  • Step 2: Move terms involving \(x\) to one side and constant terms to the other.
    In our case, after setting the function equal to 5, we have the equation: \(-2x + 5 = 5\). By subtracting 5 from both sides, we start the isolation process.
  • Step 3: Solve for \(x\) by dividing by the coefficient of \(x\).
    Here, \(-2x = 0\), and dividing both sides by \(-2\) gives \(x = 0\).
Understanding how to manipulate equations is essential for finding unknown variables and solving real-world problems.
Verification of Solutions
Verification is a vital part of solving equations, ensuring that the solution we found is indeed correct. After solving for \(x\), we should substitute it back into the original function to confirm:
  • Substitution Method: Replace the variable \(x\) in the original function with the solution value.
    Given our solution \(x = 0\), plug it into the function \(f(x) = -2x + 5\) to check if it results in the given function value of 5.
After substituting, we get \(f(0) = -2(0) + 5 = 5\). This process determines that the left side of the equation equals the right side, verifying our solution. Verification confirms the accuracy of your solution and enhances confidence that the solution is correct.

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Most popular questions from this chapter

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