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Chapter 1: Problem 118

Find \(w\) if \(f(w)=5\) and \(f(x)=-3 x-7\)

Short Answer

Expert verified
The value of w is -4.

Step by step solution

01

Understand the Function

Examine the given function: defined by \( f(x) = -3x - 7 \).It maps any input value of x to an output defined by the equation.
02

Set Up the Equation

We are given that \( f(w) = 5 \). This means we can set up the following equation by replacing f(w) with the given function:\( -3w - 7 = 5 \)
03

Isolate the Variable w

Our goal is to solve for w.First, add 7 to both sides of the equation: \( -3w - 7 + 7 = 5 + 7 \)This simplifies to: \( -3w = 12 \)
04

Solve for w

Next, we need to isolate w by dividing both sides by -3: \( w = \frac{12}{-3} \)This simplifies to \( w = -4 \)

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Key Concepts

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

Understanding Function Notation
Function notation is a way to name and define functions in mathematics. It involves using symbols like \(f(x)\) to describe a relationship between two variables.
In our example, we have the function \(f(x) = -3x - 7\). This means that for any input value \(x\), the output is calculated by the expression on the right-hand side of the equation.

To grasp function notation better:
  • The letter \(f\) represents the function's name.
  • The value inside the parentheses, \(x\), is the input of the function.
  • The equation on the right gives the rule for calculating the function's output based on the input \(x\).
In our case, if you input \(x = 2\), the function gives: \(f(2) = -3(2) - 7 = -6 - 7 = -13\). This tells us the output when the input is 2.
Isolating Variables
To solve equations, we often need to isolate a variable. This means getting the variable alone on one side of the equation. Let's break down the process step by step.

In our example, we start with the equation \(-3w - 7 = 5\). The goal is to have \(w\) on one side by itself. Here鈥檚 how:
  • Add 7 to both sides: \(-3w - 7 + 7 = 5 + 7\).
  • This simplifies to \(-3w = 12\).
The key is to use inverse operations, like adding if there's subtraction, to move terms from one side of the equation to the other. Always perform the same operation on both sides to keep the equation balanced.
Solving for a Variable
Once you've isolated the variable, the next step is to solve for it. This usually involves one last operation to get the variable by itself.

Following from our isolated variable equation \(-3w = 12\):
  • We divide both sides by \(-3\): \(w = \frac{12}{-3}\).
  • Simplify the right-hand side to get \(w = -4\).

This tells us that \(w\) must be \(-4\) to satisfy the original function equation. By isolating and solving, we鈥檝e found the value for \(w\).

It's always a good practice to check your work by plugging the solution back into the original equation to ensure it holds true.

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