Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Find the zero of the finction \(f\). Do not use a calculator. $$f(x)=3 x+6(x-4)$$

Step-by-Step Solution

Answer

The zero of the function is \(x = \frac{8}{3}\).

Step 1: Understanding the Function

The function given is composed of two terms, where the second term is a product of a constant and a linear expression: \[ f(x) = 3x + 6(x - 4) \] The task is to find the zero of this function, where the output value is zero.

Step 2: Simplify the Function

First, simplify the expression by distributing the multiplication in the second term:\[ f(x) = 3x + 6(x - 4) = 3x + 6x - 24 \]Combine like terms:\[ f(x) = 9x - 24 \]

Step 3: Set the Function Equal to Zero

To find the zero of the function, set the simplified function equal to zero:\[ 9x - 24 = 0 \]

Step 4: Solve for x

Solve the equation for \( x \) by isolating \( x \). Start by adding 24 to both sides:\[ 9x = 24 \]Now, divide both sides by 9:\[ x = \frac{24}{9} \]Simplify the fraction:\[ x = \frac{8}{3} \]

Key Concepts

Linear Equations

In algebra, a linear equation is any equation that can be written in the form \(ax + b = 0\), where \(a\) and \(b\) are constants and \(x\) is a variable. Linear equations represent straight lines when graphed on a coordinate plane.
When dealing with linear equations, it's important to understand a few key properties:

• Slope-intercept form: A common form, \(y = mx + c\), describes a line with slope \(m\) and y-intercept \(c\). In our exercise, the simplified function \(f(x) = 9x - 24\) is in a similar form with the slope of 9.
• Constant rate of change: The slope indicates a constant rate at which \(y\) changes as \(x\) changes. Here, for every unit increase in \(x\), \(f(x)\) increases by 9.
• Graphical representation: Since linear equations graph into straight lines, they are easier to visualize, making solutions like finding the zero more intuitive.

Understanding these properties can simplify the process of solving linear equations and finding zeros.

Solve for x

To solve for \(x\) means to find the value of \(x\) that makes the equation true. Consider our simplified equation from the original exercise: \[9x - 24 = 0\]Here, solving for \(x\) involves isolating \(x\) on one side of the equation. Follow these steps:

• Add any constant term on one side to both sides to remove it. So add 24 to both sides to get: \[9x = 24\]
• Next, divide both sides by the coefficient of \(x\), in this case, 9, to solve for \(x\): \[x = \frac{24}{9}\]
• Simplify the fraction, if necessary: \[x = \frac{8}{3}\] This means \(x = \frac{8}{3}\) is the solution.

Following these steps orderly ensures accuracy and understanding when solving linear equations. Regular practice can enhance proficiency in solving for variables.

Zero of a Function

The zero of a function is the value of \(x\) when the function equals zero. In other words, it's where the graph of the function intersects the x-axis.
For a function \(f(x) = ax + b\), the zero is found by setting \(f(x)\) equal to zero and solving for \(x\).
In our exercise, we are given:\[f(x) = 9x - 24\]To find the zero, set the equation equal to zero:\[9x - 24 = 0\]Solve as follows:

• Add 24 to both sides resulting in: \[9x = 24\]
• Divide both sides by 9 to solve for \(x\): \[x = \frac{24}{9}\]
• Simplify the fraction to find: \[x = \frac{8}{3}\]

Thus, the zero of the function is \(x = \frac{8}{3}\). This reflects the point \(\left(\frac{8}{3}, 0\right)\) on the graph, showing where the function crosses the x-axis.