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Chapter 2: Problem 67

If \(f(x)=4-x,\) for what input is the output \(7 ?\)

Short Answer

Expert verified
The input is \( -3 \).

Step by step solution

01

Set up the equation

Start with the function given, which is \[ f(x) = 4 - x \]We need to find the input value, let that be \( x \), for which the output \( f(x) \) is 7.
02

Substitute the output value

Substitute the given output value into the function equation. We have:\[ 7 = 4 - x \]
03

Solve for \(x\)

Rearrange the equation to solve for \( x \). Start by isolating \( x \) on one side of the equation:\[ 7 = 4 - x \]Subtract 4 from both sides:\[ 7 - 4 = - x \]This simplifies to:\[ 3 = - x \]Now, multiply both sides by -1:\[ -3 = x \]
04

Verify the solution

To ensure the solution is correct, substitute \( x = -3 \) back into the original function:\[ f(-3) = 4 - (-3) = 4 + 3 = 7 \]This confirms that the output is indeed 7 when the input is \( -3 \).

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Key Concepts

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

Function Evaluation
In mathematics, a function is like a machine that takes an input and gives an output. One common task is to determine what output a function produces for a specific input. This is known as function evaluation. For the given function, \(f(x) = 4 - x\), we plug in different values of \(x\) to see what output we get.

For example, if you wanted to find the output for \(x = 2\), you simply substitute 2 into the function equation: \(f(2) = 4 - 2 = 2\).
By understanding function evaluation, we can quickly determine outputs for any given input.
Solving for Variables
Solving for variables means finding the value of an unknown variable in an equation. In our problem, we need to find the value of \(x\) that makes the function output 7. Start by setting the function equal to 7: \(7 = 4 - x\).

Next, isolate \(x\) on one side of the equation.
  • First, subtract 4 from both sides: \(7 - 4 = - x\).
  • This simplifies to: \(3 = - x\).
  • Finally, multiply both sides by -1 to solve for \(x\): \(-3 = x\).
Now we have found the value of \(x\) which is -3.

This method of solving equations is fundamental in algebra and helps us find unknown values in various contexts.
Verification of Solutions
Verifying your solution is a crucial step in solving equations. It ensures that your answer is correct. To verify, substitute the found value back into the original equation and check if it satisfies the equation.

In this case, we found \(x = -3\). Now, substitute \(-3\) back into the function equation: \(f(-3) = 4 - (-3) = 4 + 3 = 7\).

Since the output indeed equals 7, this confirms our solution is correct.
  • Always substitute back to check.
  • Ensure no mistakes were made during calculations.
  • Verification gives confidence in your answer.
By following this final step, you can be sure that your solution is accurate and reliable.

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Most popular questions from this chapter

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge. In 2016 , for 2 nd day Zone 2 delivery of packages weighing from 10 to 50 lb, FedEx charged \(\$ 21\) plus \(\$ 1.10\) for each pound in excess of 10 lb. Estimate the weight of a package that cost \(\$ 43\) to ship.

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 8 x+40=0 $$

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ x+3=5-x $$

Write equations for two functions \(f\) and \(g\) such that the domain of \(f+g\) is \(\\{x | x \text { is a real number and } x \neq-2 \text { and } x \neq 5\\}\). Answers may vary.

Combine like terms. [ 1.3] $$a b^{2}+2 a^{2} b-a b^{2}$$
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