Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=|x-2|, x=2\)

Step-by-Step Solution

Answer

The domain of the function \(f(x) = |x - 2|\) is all real numbers, the range is \(y \geq 0\), and \(f(2) = 0\).

Step 1: Find the Domain

For any real function, the domain is the set of all real numbers for which the function is defined. Here, the function \(f(x) = |x - 2|\) is defined for all values of \(x\), because the absolute value function \(|x|\) is defined for all \(x\). So, the domain of function \(f\) is all real numbers.

Step 2: Find the Range

The output of the absolute value function \(|x|\) is always non-negative, because it represents the distance of \(x\) from zero. Similarly, the output of the function \(f(x) = |x - 2|\), which represents the distance of \(x\) from 2, will also be a non-negative real number. So, the range of the function \(f\) is \(y \geq 0\).

Step 3: Evaluate the function at \(x=2\)

Now we substitute \(x = 2\) into the function to evaluate it. So, \(f(2) = |2 - 2| = |0| = 0\).

Key Concepts

Absolute Value Function

The absolute value function is a key player in mathematics, as it measures how far a number is from zero on a number line. Think of it like measuring distance without caring about which direction you go. The absolute value of a number, represented as

• \(|x|\),

gives us the non-negative value of that number.

In the context of the function \(f(x) = |x-2|\), it does not matter if \(x\) is greater than or less than 2; what matters is the distance from 2. This is why the expression inside the absolute value, \(x-2\), changes sign depending on whether \(x\) is smaller or larger than 2, but the absolute value itself remains positive.

Key points to remember about absolute values:

• They convert any negative values to positive.
• They help calculate distances or differences without regard to direction.

Real Numbers

Real numbers encompass all the numbers that exist on the number line, which includes both rational numbers (like 1/2, 5, -3) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)).

The domain of a function describes all possible input values, and here, it consists of real numbers. For instance, when dealing with \(f(x) = |x - 2|\), there's no restriction on \(x\), meaning any real number can potentially be plugged into the function.

A few things to keep in mind about real numbers:

• They can be positive, negative, or zero.
• They can be whole numbers, fractions, or decimals.
• They represent continuous points on the number line without any gaps.

Understanding real numbers helps us explore both the domain and the range effectively, ensuring all possible outcomes and inputs are considered.

Function Evaluation

Function evaluation is about plugging a specific value of \(x\) into a function and finding the corresponding output. This is like asking, "What happens at this certain point?"

Let's take the function \(f(x) = |x - 2|\) and evaluate it at \(x = 2\). Here’s how it works:

• Replace \(x\) with the specified value: \(f(2) = |2 - 2|\).
• Simplify the expression inside the absolute value: \(f(2) = |0|\).
• Since the absolute value of zero is zero, we find \(f(2) = 0\).

This process ensures we understand not only what a function looks like but also how it behaves at particular points. Such evaluation tells us specific outputs or results the function gives under unique circumstances.