Problem 2 - A Graphical Approach to College Algebra

Question

Find the zero of the function \(f\) $$f(x)=5 x-30$$

Step-by-Step Solution

Answer

The zero of the function is \(x = 6\).

Step 1: Understand the Problem

The problem requires finding the zero of the function, which means determining the value of \(x\) that makes \(f(x) = 0\). In this context, the function given is linear: \(f(x) = 5x - 30\).

Step 2: Set the Function Equal to Zero

To find the zero of the function, set \(f(x)\) equal to zero. This gives us the equation: \(5x - 30 = 0\).

Step 3: Solve for x

Solve the equation \(5x - 30 = 0\) by first adding 30 to both sides, which results in \(5x = 30\). Then, divide both sides by 5 to isolate \(x\): \(x = \frac{30}{5} = 6\).

Step 4: Verify the Result

To ensure that \(x = 6\) is correct, substitute \(x = 6\) back into the function. Calculate \(f(6) = 5(6) - 30 = 30 - 30 = 0\). Thus, the function satisfies \(f(x) = 0\) at \(x = 6\).

Key Concepts

Linear Functions

Linear functions are some of the simplest types of functions that you will encounter in mathematics. They are defined by a straight line when graphed on a coordinate plane. The general form of a linear function is

• \(f(x) = mx + b\)

Here, \(m\) represents the slope, while \(b\) stands for the y-intercept.
Understanding these terms is crucial:

• The **slope** \(m\) indicates how steep the line is and whether it is inclining or declining.
• The **y-intercept** \(b\) is the point where the line crosses the y-axis.

In our exercise's function, \(f(x) = 5x - 30\), the slope \(m\) is 5, and the y-intercept \(b\) is -30. This means for every one unit increase in \(x\), \(f(x)\) increases by 5 units. Linear functions like this are helpful in various real-life applications where relationships between quantities are constant.

Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. For linear equations, this process is straightforward. You handle them through basic arithmetic steps.
Take our problem as an example, where we found the zero of the function \(f(x) = 5x - 30\). Finding the zero means setting the function to 0 and solving for \(x\):

• Start with the equation: \(5x - 30 = 0\).
• Add 30 to both sides to isolate terms with \(x\): \(5x = 30\).
• Divide both sides by 5 to solve for \(x\): \(x = 6\).

These simple operations reveal how useful it is to understand solving linear equations. It's all about isolating the variable on one side to find its value.

Function Verification

Once you've solved for zero, it's important to verify your solution to ensure it is correct. Verification means plugging your solution back into the original function to see if it works.
For example, after solving for \(x = 6\), substitute \(x = 6\) back into the function \(f(x) = 5x - 30\):

• Calculate \(f(6) = 5(6) - 30\).
• Simplify to \(30 - 30 = 0\).

The result is indeed 0, confirming that your solution is correct. This step not only affirms your answer but helps in understanding how changes in variables affect the function. Verification is a critical step in problem-solving as it validates the accuracy of your solution.