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Chapter 1: Problem 41

Find the value of \(x\) in the domain of \(f(x)=1-4 x\) for which \(f(x)=3\).

Short Answer

Expert verified
The value of \(x\) for which \(f(x)=3\) in the function \(f(x)=1-4 x\) is \(x=-0.5\).

Step by step solution

01

Equate Function to Given Value

Set the given function \(f(x) = 1 - 4x\) equal to 3. This gives the equation \(1 - 4x = 3\).
02

Simplify Equation

Perform the arithmetic operation on the left side of the equation to isolate the term with the variable \(x\). Here you subtract 1 from both sides of the equation to yield \(-4x = 3 - 1\). This simplifies to \(-4x = 2\).
03

Solve for Variable

Solve the simplified equation \(-4x = 2\) for \(x\) by dividing both sides of the equation by -4. This yields \(x = 2/-4 = -0.5\).

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Key Concepts

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

Function Domain
A function's domain is all the possible values that a variable, often "\(x\)", can take. For the function \(f(x) = 1 - 4x\), the variable \(x\) can be any real number. This means there are no restrictions such as square roots of negative numbers or division by zero that would limit \(x\).
Understanding the domain is crucial because it tells us the possible input values that lead to valid outputs in the function.
For different functions, always check for expressions like:\
    \
  • Square roots: Avoid negatives inside the root unless in complex numbers.
    • \
  • Fractions: Ensure the denominator is not zero.
    • \
  • Logarithms: Only positive numbers are allowed inside a log function.
    • \
For our exercise, since \(f(x) = 1 - 4x\) is a simple linear equation, \(x\) has no restrictions.
Arithmetic Operations
Arithmetic operations involve the basic mathematical operations: addition, subtraction, multiplication, and division.
In the exercise, we employed subtraction and division to simplify the equation. Initially, the equation given was \(1 - 4x = 3\), where subtraction is used to rearrange and simplify.
    \
  • Subtract 1 from both sides: This operation helps in isolating the term with the variable. It transforms \(1 - 4x = 3\) to \(-4x = 2\).
    • \
These operations are fundamental in manipulating equations to isolate variables and find their values.
Variable Isolation
Variable isolation is the process of manipulating the equation so that the unknown variable is alone on one side of the equation. This makes it easier to solve for the variable.
In our problem, we want to find \(x\), so we manipulate the equation \(-4x = 2\) by performing division.
    \
  • Divide each side by -4: Doing this isolates \(x\) on one side, resulting in \(x = -0.5\).
    • \
Isolating the variable is a common and essential strategy in algebra, especially when working with linear equations.
This step ensures clarity and precision, helping us find the correct value for \(x\) in the equation.
Solving Equations
Solving equations involves finding the value of the unknown variable that makes the equation true.
In the exercise provided, we began with setting up \(1 - 4x = 3\). The main objective is to manipulate the equation to reveal the value of \(x\).
Steps for solving include:
    \
  • Rearrange the equation: Use subtraction and division to simplify.
    • \
  • Isolate the variable: \(-4x = 2\) is solved by dividing both sides by -4 to achieve \(x = -0.5\).
    • \
By following these steps, we confidently find that the solution for \(x\) is \(-0.5\).
Mastering solving techniques is vital not only for linear equations but for solving many other kinds of equations in mathematics.

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Most popular questions from this chapter

A boat was purchased for \(\$ 44,000 .\) Assuming the boat depreciates at a rate of \(\$ 4200\) per year (straight-line depreciation) for the first 8 years, write the value \(v\) of the boat as a function of the time \(t\) (measured in years) for \(0 \leq t \leq 8\).

Use interval notation to express the solution set of each inequality. $$|2 x+7| \leq 0$$

Use interval notation to express the solution set of each inequality. $$|x-3|<10$$

The number of diagonals \(D\) of a polygon with \(n\) sides is given by the formula $$D=\frac{n}{2}(n-3)$$ a. Determine the number of sides of a polygon with 464 diagonals. b. Can a polygon have 12 diagonals? Explain.

Solve by completing the square or by using the quadratic formula. $$2 x^{2}+\sqrt{5} x-3=0$$
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