Problem 10 - College Algebra

Question

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=\sqrt{x}\) and \(g(x)=|x-3|,\) find \(\frac{g}{f}\)

Step-by-Step Solution

Answer

The domain of \( \frac{g}{f} \) is \( (0, \infty) \).

Step 1: Determine the Domain of f(x)

The function is given as \( f(x) = \sqrt{x} \). For the square root to be defined and real, the expression inside the square root must be non-negative. Therefore, \( x \geq 0 \). Thus, the domain of \( f(x) \) is \( [0, \infty) \).

Step 2: Determine the Domain of g(x)

The function \( g(x) = |x-3| \) is an absolute value function, which is defined for all real numbers. Hence, the domain of \( g(x) \) is \( (-\infty, \infty) \).

Step 3: Determine the Domain of \( \frac{g}{f} \)

The function \( \frac{g}{f} = \frac{|x-3|}{\sqrt{x}} \) is a quotient of two functions. The domain is determined by both functions’ domains and ensuring the denominator is not zero. Since \( f(x) \) must be non-zero, we set \( \sqrt{x} eq 0 \), which implies \( x > 0 \). Considering the domain restrictions, the real domain of \( \frac{g}{f} \) is \( (0, \infty) \).

Step 4: Express the Domain in Interval Notation

Combining the restrictions from the previous step, the final domain of the function \( \frac{g}{f} \) in interval notation is \( (0, \infty) \).

Key Concepts

Interval Notation

Interval notation is a fantastic way to describe the set of values that belong to the domain of a function. In interval notation, we use brackets to tell us which numbers are included or excluded. For example:

• Square brackets, like \([a, b]\), mean that both endpoints \(a\) and \(b\) are included in the set of numbers.
• Parentheses, such as \((a, b)\), exclude the endpoints \(a\) and \(b\).
• Combinations like \((a, b]\) mean \(a\) is excluded, but \(b\) is included, and vice versa for \([a, b)\).

For the function \(f(x) = \sqrt{x}\), the domain is restricted to non-negative numbers because square roots of negative numbers aren't real. This is why we use \([0, \infty)\). The square bracket on 0 shows 0 is included, indicating that the function starts at zero and extends to infinity. On the other hand, a function with no restrictions like the absolute value function \(g(x) \) covers all real numbers, expressed as \((-\infty, \infty)\). Thus, for a composite function like \(\frac{g}{f}\), checking both functions' domains and preventing division by zero, the domain becomes \((0, \infty)\), where parentheses are due to excluding zero itself, as we don't divide by zero.

Absolute Value Function

The absolute value function might sound fancy, but it’s really simple. It’s written as \(|x|\) and gives the magnitude or distance of a number from zero. No matter if \(x\) is positive or negative, the result is always non-negative. This is why:

• For positive numbers, say \(x = 4\), the absolute value is \(|4| = 4\).
• For negative numbers, like \(x = -4\), the absolute value flips the sign: \(|-4| = 4\).
• At zero, \(|0| = 0\).

The function \(g(x) = |x-3|\) shifts the basic \(x\)-axis centered function to the right by 3 units. It is always defined since it can handle all real numbers. This is because, in simple terms, absolute value adjusts the negative inputs to their positive equivalents but doesn’t touch positive ones. This gives it the entire set of real numbers as a valid domain, denoted as \((-\infty, \infty)\) in interval notation.

Square Root Function

A square root function is a type of expression, like \(f(x) = \sqrt{x}\), that takes the square root of its input. The result of a square root is always non-negative, leading to an important limitation - the input must also be non-negative for real numbers. That's why the domain normally looks like this:

• It starts from zero and upwards, shown as \([0, \infty)\) in interval notation.

Zero is the smallest number we can plug in because the square root of 0 is perfectly fine and real, equal to 0. However, any negative number would make the expression complex, thus not part of the real number family. When working with more complex functions like \(\frac{g}{f} = \frac{|x-3|}{\sqrt{x}}\), the denominator \(\sqrt{x}\) requires special attention. We exclude zero, \((0, \infty)\), to ensure we're not dividing by zero. This simple understanding of a square root function’s domain is crucial when dealing with more complicated mix-ups with other functions.