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Chapter 1: Problem 4

Find the zero of the function \(f\) $$f(x)=-2 x$$

Short Answer

Expert verified
The zero of the function is \(x = 0\).

Step by step solution

01

Set the Function to Zero

To find the zero of the function, set the function equal to zero: \[ f(x) = -2x = 0 \]
02

Solve for x

Solve the equation from Step 1 for \(x\). Since \(-2x = 0\):\[ x = \frac{0}{-2} = 0 \]
03

Verify the Solution

Verify that the solution \(x = 0\) satisfies the original equation.Substitute \(x = 0\) back into the function:\[ f(0) = -2(0) = 0 \]Since this is true, \(x = 0\) is indeed the zero of the function.

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Key Concepts

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

Linear Equations
Linear equations are mathematical expressions that represent a straight line when plotted on a graph. They are the simplest type of polynomial equations and are usually written in the form:
\[ ax + b = 0 \]
where \(a\) and \(b\) are constants, and \(x\) is the variable or unknown value we are trying to solve.
  • In a linear equation, \(a\) is the coefficient of \(x\), determining the slope of the line.
  • \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
  • The goal is to find the value of \(x\) that makes the equation true.
A key property of linear equations is that they have a single solution when they are non-degenerate (\(a eq 0\)). This solution will give the x-coordinate of the point where the line crosses the x-axis, also known as the zero or root of the function. For the example given, the equation \(-2x = 0\) is a linear equation with no \(b\) value, making it even simpler to solve.
Solving Equations
Solving equations involves finding the value of the variable that makes the entire statement true. For linear equations, this process is straightforward:
  • First, set the equation equal to zero. This is known as finding the root or zero of the function. For instance, setting \(-2x = 0\).
  • Next, isolate the variable \(x\) by performing arithmetic operations. Here, dividing both sides of the equation by -2: \[ x = \frac{0}{-2} = 0 \].
These steps can include:
  1. Adding or subtracting constants from both sides to eliminate any additional terms.
  2. Dividing or multiplying to clear coefficients.
The aim is to ensure \(x\) is by itself on one side of the equation, providing the solution to the problem. It's important because it represents a valid point where the function intersects the x-axis, effectively being the x-value for which the function's output is zero.
Verifying Solutions
Verifying solutions in solving equations is an essential step in the process. It ensures the obtained solution is correct by substituting it back into the original equation.
  • After solving \(x = 0\), substitute it into the original function \(f(x) = -2x\): \[ f(0) = -2(0) = 0 \].
  • Check if the result matches your expectation (in this case, zero).
This step involves plugging the solution back into the equation to confirm that both sides of the equation are indeed equal. If they are equal, then the solution is verified as correct.
Verification is crucial, especially when dealing with more complex equations, as it serves as a final check. This simple practice helps prevent errors and provides confidence in the solution process, confirming that every calculation step was accurate.

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