Problem 9 - A Graphical Approach to College Algebra

Question

Find the zero of the function \(f\) $$f(x)=1.5 x+2(x-3)+5.5(x+9)$$

Step-by-Step Solution

Answer

The zero of the function is at \(x = -4.8333\).

Step 1: Simplify the Function

First, distribute the terms in the function.By expanding:\(f(x) = 1.5x + 2(x - 3) + 5.5(x + 9)\)\(= 1.5x + 2x - 6 + 5.5x + 49.5\)Combine like terms:\(f(x) = (1.5x + 2x + 5.5x) + (-6 + 49.5)\)Simplify:\(f(x) = 9x + 43.5\)

Step 2: Set the Function to Zero

To find the zero of the function, set the simplified function equal to zero:\(9x + 43.5 = 0\)

Step 3: Solve for x

Subtract 43.5 from both sides to isolate terms with \(x\):\(9x = -43.5\)Divide both sides by 9:\(x = \frac{-43.5}{9}\)Simplify the fraction:\(x = -4.8333\)

Key Concepts

Algebraic Simplification

Algebraic simplification is an essential skill in mathematics that makes working with complex expressions more manageable. It involves reducing expressions to their simplest form by applying basic algebraic rules. In the given function, simplification starts by expanding the expression:\[f(x) = 1.5x + 2(x - 3) + 5.5(x + 9)\]This involves distributing the multiplication over addition or subtraction to simplify each term individually.Benefits of algebraic simplification include:

• Making calculations more straightforward and less prone to errors.
• Revealing the underlying structure of an expression, which can help in further manipulations or problem-solving.
• Facilitating easier identifications of zeros, maxima, minima, or other properties of functions.

Simplifying expressions is often the first step when tackling algebraic problems, making it a crucial concept to understand and master.

Linear Equations

Linear equations represent a fundamental concept in algebra where the highest power of the variable is one. These equations take the form \(ax + b = 0\), where \(a\) and \(b\) are constants, and \(x\) is the variable. In the problem, the function simplifies to a linear form involving a single variable:\[f(x) = 9x + 43.5\]When finding zeros of a function, you set the linear equation equal to zero and solve for the variable. This process essentially finds the point where the function intersects the x-axis. Here is how it proceeds:

• Set the linear equation to zero: \(9x + 43.5 = 0\)
• Solve for \(x\) using algebraic manipulations such as subtraction and division.
• This gives \(x = -4.8333\)

Understanding linear equations is vital as they form the basis for many real-world applications, such as calculating distances, predicting outcomes, and optimizing functions.

Function Expansion

Function expansion involves expressing a function in an extended form to reveal its individual components. This process helps in better understanding and manipulation of the function. In our example, the expression\[f(x) = 1.5x + 2(x - 3) + 5.5(x + 9)\]is expanded by multiplying each set of terms:

• \(2(x - 3)\) becomes \(2x - 6\)
• \(5.5(x + 9)\) becomes \(5.5x + 49.5\)

Each group of terms is broken down to understand their contribution to the overall function. Expanding a function is often used in calculus and physics to simplify derivatives and integrals. It can also help in identifying patterns or simplifying further operations.

Combining Like Terms

After expanding a function, the next logical step is to combine like terms. These are terms in an expression that have the same variable part. By combining them, we further simplify the expression into a more interpretable form. For example, once the function was expanded:\[f(x) = 1.5x + 2x - 6 + 5.5x + 49.5\]We can simplify it by combining like terms:

• Combine all the terms involving \(x\): \(1.5x + 2x + 5.5x = 9x\)
• Combine the constant terms: \(-6 + 49.5 = 43.5\)

Thus, the function simplifies to: \(f(x) = 9x + 43.5\). Combining like terms is a fundamental process that ensures our expressions remain as simple as possible, making subsequent calculations or evaluations much easier.