Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Find the zero of the function \(f\). \(f(x)=-8 x+0.5(2 x+8)\)

Step-by-Step Solution

Answer

The zero of the function is \(x = \frac{4}{7}\).

Step 1: Distribute the Constants

Expand the expression by distributing the constant over the terms inside the parentheses. This means multiplying 0.5 with each term inside the parentheses: \[f(x) = -8x + 0.5(2x) + 0.5(8)\]This simplifies to:\[f(x) = -8x + x + 4\]

Step 2: Combine Like Terms

Combine the terms that have the variable 'x'. In this case, combine \(-8x\) and \(x\).\[-8x + x = -7x\]Substitute it back into the function:\[f(x) = -7x + 4\]

Step 3: Set the Function to Zero

To find the zero of the function, set \(f(x) = 0\) and solve for \(x\).\[-7x + 4 = 0\]

Step 4: Solve for x

Isolate \(x\) to solve the equation:Subtract 4 from both sides:\[-7x = -4\]Divide each side by -7:\[x = \frac{-4}{-7}\]This simplifies to:\[x = \frac{4}{7}\]

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle that allows us to simplify expressions by distributing a number over addition or subtraction within parentheses. Imagine you have a basket with apples and oranges, and you want to share them equally among friends. Similarly, when you apply the distributive property, you are sharing a multiplying number with every term inside parentheses.
To illustrate, let’s take our example function:

• Original function: \(f(x) = -8x + 0.5(2x + 8)\).
• Applying the distributive property means multiplying 0.5 by both terms inside the parentheses: \(0.5 imes 2x\) and \(0.5 imes 8\).
• This results in: \(f(x) = -8x + x + 4\).

The distributive property helps in breaking down and expanding expressions, setting the stage for easier problem-solving and manipulation in algebra.

Combining Like Terms

After using the distributive property, the next essential step in simplifying an expression is combining like terms. Like terms are terms that contain the same variables raised to the same power, only differing by their coefficients. The idea is much like gathering all your apples in one basket and oranges in another, ensuring that each type of fruit is easy to count.
In the expression from our example:

• We have: \(-8x\) and \(+x\).
• Both terms have the variable \(x\), making them like terms.
• To combine them, simply perform the arithmetic operation: \(-8x + x = -7x\).

This simplification results in a cleaner and more manageable expression: \(f(x) = -7x + 4\). Combining like terms is critical for arriving at a simpler and more easily solvable expression.

Linear Equations

Linear equations are fundamental to understanding algebra as they help us determine the relationship between variables. A linear equation typically forms a straight line when graphed and can be presented in the form \(ax + b = 0\). The zeros or roots of a linear equation are the values of \(x\) that make the equation equal to zero.
For the function in our example:

• We set the equation \(-7x + 4 = 0\) to find the zero of the function.
• To find \(x\), subtract 4 from both sides: \(-7x = -4\).
• Divide each side by \(-7\) to isolate \(x\): \(x = \frac{4}{7}\).

This provides the zero of the function. Understanding linear equations is crucial because solving these equations is a key step in finding where the function equals zero, which is an essential concept in many applications of math and science.