Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Find the zero of the function \(f .\) Do not use a calculator. \(f(x)=-2 x\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are asked to find the zero of the function \(f(x) = -2x\). A zero of a function is any \(x\) value such that \(f(x) = 0\).

Step 2: Setting the Function to Zero

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

Step 3: Solving for x

To find the value of \(x\) that makes \(-2x = 0\) true, divide both sides of the equation by \(-2\): \[ x = \frac{0}{-2} \] This simplifies to \(x = 0\).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of mathematical functions. They form a straight line when graphed. A linear function can generally be written in the form \(f(x) = ax + b\). Here, \(a\) and \(b\) are constants. This particular structure is essential because it allows us to determine the behavior of the function quickly.
For example:

• The coefficient \(a\) is the slope of the line. It tells us how steep the line is and in which direction it slants.
• The constant \(b\) is the y-intercept, where the line crosses the y-axis.

In our problem, the function is \(f(x) = -2x\). It is a special type where \(b = 0\), meaning the line passes through the origin, \((0,0)\). This kind of linear function is known as a pure linear function, as it directly multiplies the input \(x\) by the constant \(a\), in this case, \(-2\). This makes the computation straightforward and intuitive.

Equation Solving

Equation solving is the process of finding the value that makes an equation true. Here, our function \(f(x) = -2x\) leads us to the equation \(-2x = 0\). Our task is to solve this equation, which involves isolating the unknown variable \(x\).
To find the solution:

• First, identify the terms on each side of the equation. In this instance, \(-2x\) and \(0\).
• Next, simplify the process by trying to isolate the variable \(x\). This is done by performing operations that maintain the equality, such as addition, subtraction, multiplication, and division.

Since the only term with \(x\) is \(-2x\), dividing both sides by \(-2\) removes the coefficient and solves for \(x\). Importantly, division is valid here because \(-2\) is not zero, ensuring the operation's validity. In the end, the solution \(x = 0\) satisfies this simple equation.

Basic Algebra

Basic algebra involves understanding how to manipulate numbers and symbols to express relationships between quantities. In finding the zero of a function, like in our problem \(f(x) = -2x\), algebra becomes useful in restructuring and simplifying mathematical expressions.
This exercise demonstrates:

• The use of setting equations to zero to find unknowns, a core algebraic concept.
• Breaking down equations by performing inverse operations, such as dividing both sides by a constant. This technique simplifies the equation step-by-step.

In basic algebra, zero often plays a crucial role since establishing a thing to have the value zero can simplify equations and expressions. Understanding zeros of functions enables the prediction of behaviors of more complex mathematical models. By mastering algebra, students lay the foundation for understanding more advanced mathematics concepts later on.