Problem 10 - College Algebra with Corequisite Support

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: Understand Function Requirements

The function \( f(x) = \sqrt{x} \) requires the input \( x \) to be non-negative because the square root of a negative number is not real. Hence, the domain of \( f(x) \) is \( x \geq 0 \).

Step 2: Analyze Domain of g(x)

The function \( g(x) = |x - 3| \) is defined for all real numbers, hence the domain of \( g(x) \) is \( (-\infty, \infty) \).

Step 3: Formulate \(\frac{g}{f}\)

The expression \( \frac{g}{f}(x) = \frac{|x-3|}{\sqrt{x}} \) is formed by dividing \( g(x) = |x - 3| \) by \( f(x) = \sqrt{x} \).

Step 4: Determine Combined Domain

To determine the domain of \( \frac{g}{f} \), we need \( \sqrt{x} eq 0 \), leading to \( x > 0 \). Since \( x \geq 0 \) for \( f(x) \), excluding \( x = 0 \), the domain is \( (0, \infty) \).

Step 5: Conclude Domain in Interval Notation

Thus, the domain of \( \frac{g}{f} \) in interval notation is \( (0, \infty) \). This reflects the requirement that \( f(x) \) must be positive and as \( \frac{g(x)}{f(x)} \) cannot be evaluated at \( x = 0 \) without becoming undefined.

Key Concepts

Interval Notation

Interval notation is a simple way to describe a set of numbers between two endpoints. It uses brackets and parentheses to indicate whether the endpoints are included or excluded in the range.

There are two types of brackets:

• Square brackets [ ] indicate that the endpoint is included in the interval (closed interval).
• Parentheses ( ) imply that the endpoint is not included in the interval (open interval).

For example, the interval \( [a, b) \) refers to all real numbers greater than or equal to \( a \) and less than \( b \). This means \( a \) is included, but \( b \) is not. On the contrary, \( (a, b] \) includes \( b \, \) but not \( a \).

This notation efficiently communicates the domain of functions, as seen in our exercise with the domain \( (0, \, \infty) \), indicating all real numbers greater than 0.

Square Root Function

The square root function, denoted as \( \sqrt{x} \, \) is a fundamental mathematical function that relates any non-negative number \( x \) to a number that, when squared, yields \( x \. \) The function \( f(x) = \sqrt{x} \) places certain requirements on its input, namely that it must not be negative.

The square root of negative numbers is not a real number, which restricts the domain of \( f(x) = \sqrt{x} \, \) to all real numbers greater than or equal to zero (i.e., \( x \geq 0 \)). This is why in our exercise, the domain for the function \( f \) is expressed as \( [0, \infty) \).

Square root functions are versatile and can be seen in various mathematical contexts. Their graphs typically start at the origin and extend infinitely to the right, illustrating the non-negative domain.

Absolute Value Function

The absolute value function \( g(x) = |x-3| \) showcases a unique behavior—it measures the distance of a number from zero on the number line, effectively ignoring negative signs. This function is defined for all real numbers, resulting in a domain of \((-\infty, \infty)\).

For any number \( x \), \(|x-3|\) should be interpreted as the magnitude of the difference between \( x \) and 3 without regard to direction.

When plotting an absolute value function, you'll notice it forms a "V" shape graph, with its lowest point, or "vertex", occurring where the inside expression equals zero. This property makes absolute values significant in problems involving distance and magnitudes. The versatility and comprehensive domain of absolute values make them useful in numerous mathematical scenarios.