Problem 1 - A Graphical Approach to College Algebra

Question

Find the zero of the function \(f\) $$f(x)=-3 x-12$$

Step-by-Step Solution

Answer

The zero of the function is at \(x = -4\).

Step 1: Understand the Problem

We need to find the zero of the function, which means solving for the value of \(x\) such that \(f(x) = 0\). Given function is \(f(x) = -3x - 12\).

Step 2: Set the Function Equal to Zero

To find the zero of the function \(f(x)\), we set the equation \(-3x - 12 = 0\).

Step 3: Solve for \(x\)

We solve the equation \(-3x - 12 = 0\) by isolating \(x\): 1. Add 12 to both sides: \(-3x - 12 + 12 = 0 + 12\) This simplifies to \(-3x = 12\).2. Divide both sides by \(-3\): \(x = \frac{12}{-3}\). This simplifies to \(x = -4\).

Step 4: Verify the Solution

Substitute \(x = -4\) back into the original function to ensure it equals zero:\(f(-4) = -3(-4) - 12 = 12 - 12 = 0\),which confirms \(x = -4\) is correct.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and involve expressions where each term is either a constant or a product of a constant and a single variable. In its simplest form, a linear equation can be written as:

• Ax + B = 0, where A and B are constants, and x is the variable.

These equations graph as straight lines on a coordinate plane.
To find solutions for linear equations involves isolating x, which means "solving for x." If you have a linear function, like in our exercise with the function defined as \[ f(x) = -3x - 12 \],you should set up the equation such that
the left-hand side equals zero to find its root or zero.In this case, you solve for x to find when \(-3x - 12 = 0\).Once you solve the equation as given, you typically visualize it as a straight line crossing through the x-axis.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true.

• Start by simplifying both sides if needed.
• Use inverse operations to isolate the variable.

With our equation \(-3x - 12 = 0\),we simplify by moving constants to the other side. Bringing -12 across the equals sign involves adding 12 to both sides, to give:\[-3x = 12\].Next, to isolate x, we perform the inverse of multiplication, which is division. Divide both sides by \(-3\),and we find:\[x = \frac{12}{-3} = -4\].This process allows us to see that the solution for x where the function equals zero is \(-4\).

Function Notation

Function notation is a way to describe mathematical situations with functions.
Using symbols like \(f(x)\), function notation allows us to express the relationship between input \(x\) and output.

• The function \(f\) is applied to \(x\), hence the expression \(f(x)\).
• Here, the function described is \(f(x)=-3x-12\).

This notation clarifies that \(x\) yields a result output by the rule \(-3x-12\).
Finding the zeros of functions means identifying the input values where the output is zero. Set the equation \(f(x)=0\) to solve it, as has been demonstrated.
Remember that understanding function notation is key in math because it notates which value is substituted in and what results from given operations throughout the function's expression.