Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Match the function with its exact number of zeros. $$f(x)=x-14$$ (a) 1 zero (b) 3 zeros (c) 4 zeros (d) 5 zeros

Step-by-Step Solution

Answer

The function \(f(x) = x - 14\) has 1 zero, so the correct answer is (a) 1 zero.

Step 1: Understand the function

The function \(f(x) = x - 14\) is a linear function. A linear function is of the form \(f(x) = mx + c\), where m is the slope and c is the y-intercept.

Step 2: Set the function equal to zero

To find the zero(es) of a function, you set the function equal to zero. So we get the equation \(x - 14 = 0\), which we will solve for x.

Step 3: Solve the equation

Solving for x is straightforward, we get \(x = 14\).

Step 4: Determine the number of zeros

The function equals zero at exactly one point: \(x = 14\). Therefore, the function has 1 zero.

Key Concepts

Solving Linear Equations

Linear equations are equations of the first degree, which means they have variables raised only to the power of one and do not contain any products or roots of variables. The general form of a linear equation is expressed as:

• \(ax + b = 0\) where \(a\) and \(b\) are constants.

To solve linear equations, you aim to isolate the variable on one side of the equation. This process includes:

• Combining like terms, if necessary.
• Adding or subtracting terms from both sides.
• Multiplying or dividing both sides by a constant to solve for the variable.

For example, solving \(x - 14 = 0\) involves adding 14 to both sides, resulting in \(x = 14\). This solution tells us the value of \(x\) that satisfies the equation.

Finding Zeros of a Function

Zeros of a function are the points where the graph of the function crosses the x-axis. These points are also known as roots or x-intercepts. To find the zeros, you:

• Set the function equal to zero.
• Solve the resulting equation.

For linear functions like \(f(x) = x - 14\), you can easily set \(f(x) = 0\) to find the zero. This translates to solving the equation \(x - 14 = 0\), which gives you \(x = 14\) as the zero of the function. There is one zero because a linear function can cross the x-axis at most once.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. An example of an algebraic expression is \(x - 14\). Key concepts include:

• Variables: symbols used to represent unspecified numbers. In \(x - 14\), \(x\) is the variable.
• Constants: specific numbers. In \(x - 14\), 14 is the constant.
• Terms: parts of the expression separated by plus or minus signs.

In algebraic expressions, operations are performed according to the rules of arithmetic and algebra. The goal is often to simplify these expressions or evaluate them for given values of variables, much like determining the solution for equations.