Problem 23 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(f(x)=\frac{x}{|x|}\) a. \(f(6)\) b. \(f(-6)\)

Step-by-Step Solution

Answer

The value of the function \( f(x)=\frac{x}{|x|} \) at x=6 is 1 and at x=-6 is -1.

Step 1: Evaluate for positive x

Let's start by evaluating the function at the positive x-value, which in this case is 6. Plugging 6 into our function gives us \(f(6) = \frac{6}{|6|}\). Because 6 is positive, its absolute value remains the same, so we obtain \(f(6) = \frac{6}{6} = 1.\)

Step 2: Evaluate for negative x

Next, let's evaluate the function at the negative x-value, which in this case is -6. Plugging -6 into our function produces \(f(-6) =\frac{-6}{|-6|}\). The absolute value of -6 is 6, so the denominator becomes 6. This leads to \(f(-6) =\frac{-6}{6} = -1.\)

Key Concepts

Understanding Absolute Value

The absolute value of a number is essentially its distance from zero on a number line, without considering its direction. This means that no matter whether the number is positive or negative, its absolute value is always a non-negative number.

For example:

• The absolute value of 6 is 6, written as \(|6| = 6\).
• Similarly, the absolute value of -6 is 6, written as \(|-6| = 6\).

Absolute values are crucial when evaluating functions, especially when dealing with expressions that include both positive and negative numbers. By turning negatives into positives, the absolute value ensures we focus on magnitude rather than direction.

Function Evaluation Essentials

Function evaluation is the process of finding the output of a function for a specific input value. It involves substituting the given value into the function for the variable and performing the necessary calculations.

To evaluate a function, follow these steps:

• Identify the variable in the function. In our example, the variable is \(x\) in \(f(x) = \frac{x}{|x|}\).
• Substitute the given value into the function. For instance, when evaluating \(f(6)\), replace \(x\) with 6.
• Calculate the expression using the appropriate mathematical operations such as division or absolute value calculation.

Function evaluation is a key skill in algebra, allowing you to determine outputs and understand how changing inputs affect your results.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They form the foundation of algebra and help us generalize mathematical relationships. In our example, \(f(x) = \frac{x}{|x|}\) is an expression that involves:

• Variables: such as \(x\), which can take on different values.
• Mathematical operations: like division and absolute value.

When dealing with algebraic expressions:

• Always pay attention to operations and their precedence. This includes special operations like absolute values which affect the sign of their operands.
• Understand that expressions can change with different input values, altering the relationship you initially observe.

Through algebraic expressions, we can model real-world situations and solve complex problems by manipulating variables and numbers in a systematic way.