Problem 27 - Precalculus

Question

Use the given function \(f\) to find \(f(0)\) and solve \(f(x)=0\) $$f(x)=2 x-1$$

Step-by-Step Solution

Answer

\(f(0) = -1\) and \(x = \frac{1}{2}\) for \(f(x) = 0\).

Step 1: Evaluate f(0)

To find the value of the function at 0, simply substitute 0 for \(x\) in the function \(f(x) = 2x - 1\). Calculate as follows: \(f(0) = 2(0) - 1 = -1\). Thus, \(f(0) = -1\).

Step 2: Solve f(x) = 0

To find \(x\) where \(f(x) = 0\), set the function equal to zero and solve for \(x\). Start with the equation \(2x - 1 = 0\).

Step 3: Isolate x

Add 1 to both sides of the equation: \(2x - 1 + 1 = 0 + 1\), which simplifies to \(2x = 1\).

Step 4: Solve for x

Divide both sides by 2 to isolate \(x\): \(x = \frac{1}{2}\). Therefore, the solution to \(f(x) = 0\) is \(x = \frac{1}{2}\).

Key Concepts

Function Evaluation

Function evaluation is an important concept that allows us to determine the output of a function for a specific input. To evaluate a function, you substitute the given value into the function's formula, and then perform the arithmetic operations. For example, if you have the function \(f(x) = 2x - 1\), and you want to find \(f(0)\), you replace every instance of \(x\) with 0.

• Substitute: \(f(0) = 2(0) - 1\)
  
• Simplify: \(f(0) = 0 - 1 = -1\)
  

Here, \(f(0)\) is calculated to be -1. This process is the same regardless of what value you are substituting in for \(x\). It's like turning the function's formula into a simple equation that you can solve step by step. Evaluating functions helps us understand how inputs are transformed into outputs, and is a foundational skill in algebra and calculus.

Solving Equations

Solving equations is like cracking a puzzle, where you're finding the mystery number that makes the equation true. For a linear function like \(f(x) = 2x - 1\), solving the equation \(f(x) = 0\) helps to find where the function hits the x-axis, known as the "root" or "zero" of the function.
You start by setting the function equal to zero, making it into a solvable equation:

• Write: \(2x - 1 = 0\)
  
• Add 1 to both sides to start isolating \(x\): \(2x = 1\)
  
• Finally, divide by 2 to solve for \(x\): \(x = \frac{1}{2}\)
  

Therefore, the solution, \(x = \frac{1}{2}\), tells us that when \(x\) is \(\frac{1}{2}\), the function's output, \(f(x)\), will be zero. Solving these linear equations involves straightforward manipulation aimed at isolating \(x\), and mastering this technique is crucial for more complex algebraic manipulations.

Function Notation

Function notation is a way of representing functions in algebra. It's like a naming system that clearly shows the relationship between variables. In function notation, \(f(x)\) is the standard format and it signifies a function, \(f\), with \(x\) as its input. In the expression \(f(x) = 2x - 1\), "\(f\)" names the function while "\(x\)" indicates the input variable. Think of \(f(x)\) as a machine that takes the input \(x\), performs operations according to its rules \(2x - 1\), and gives an output.

• "f" is the function name, it could be any letter.
  
• "x" is a placeholder that represents input values.
  

Function notation ensures that when we're working with multiple functions, each can be clearly distinguished by its notation. It's a core part of how we communicate and work with functions in mathematics.