Problem 57 - Algebra and Trigonometry

Question

Find the domain of the function. $$g(x)=\frac{\sqrt{2+x}}{3-x}$$

Step-by-Step Solution

Answer

The domain is \([-2, 3) \cup (3, \infty)\).

Step 1: Understand Domain Constraints for Square Root

The domain of a function consists of all the possible input values (x-values) for which the function is defined. For the square root function \(\sqrt{2+x}\) to be defined, its radicand (the expression under the square root) must be greater than or equal to zero. This gives us the inequality: \(2 + x \geq 0\). Solve this inequality to find: \(x \geq -2\).

Step 2: Identify Domain Constraints for Denominator

The denominator of a rational function cannot be zero, as division by zero is undefined. For the function \(g(x) = \frac{\sqrt{2+x}}{3-x}\), the denominator is \(3-x\). Set \(3-x eq 0\), which simplifies to \(x eq 3\).

Step 3: Combine Conditions

The domain of the function will be all x-values that satisfy both conditions: \(x \geq -2\) and \(x eq 3\). Therefore, the domain in interval notation is \([-2, 3) \cup (3, \infty)\).

Step 4: Write Final Domain in Interval Notation

After considering both constraints, write the domain for the function \(g(x)\) as an interval. Thus, the domain in interval notation is \([-2, 3) \cup (3, \infty)\), meaning that \(x\) can take any value between \(-2\) and \infty, except at \(x = 3\).

Key Concepts

Rational Function

Let's start by understanding what a rational function is. A rational function is essentially a fraction where both the numerator and the denominator are polynomials. The general form for a rational function is \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\) because division by zero is undefined. The function we're examining in the exercise, \(g(x) = \frac{\sqrt{2+x}}{3-x}\), is a rational function because it has a polynomial in the denominator, specifically \(3-x\). When dealing with rational functions, the domain is determined by finding out which x-values make the denominator zero and excluding those from the set of possible inputs. In this case, for \(3-x = 0\), we find that \(x = 3\), so \(3\) is excluded from the domain of \(g(x)\). Understanding rational functions is crucial for determining their domains. Always watch out for those values that result in a zero denominator.

Square Root Function

The square root function is one of the simplest and most studied radical functions. A square root function looks like \(\sqrt{b+x}\) and is defined only when the expression inside the square root is non-negative. This is because you cannot take the square root of a negative number in the realm of real numbers. So, for a square root function, the expression underneath the square root, known as the radicand, must be at least zero. In the function \(g(x) = \frac{\sqrt{2+x}}{3-x}\), the square root function is \(\sqrt{2+x}\). To ensure that \(\sqrt{2+x}\) is defined, we set up an inequality: \(2 + x \geq 0\). Solving this gives \(x \geq -2\), which tells us that x must be greater than or equal to \(-2\) for the square root to be a real number. This step of identifying where a square root function is defined is key in determining the domain of a function that includes a square root.

Interval Notation

Interval notation is a way of writing subsets of the real number line, which is especially handy when discussing domains of functions. It uses parentheses \(()\) and brackets \([]\) to show where a set of numbers begins and ends, with parentheses indicating that an endpoint is not included, and brackets indicating that it is included.For the function \(g(x) = \frac{\sqrt{2+x}}{3-x}\), from our earlier work, we know that \(x\) needs to be greater than or equal to \(-2\) and not equal to \(3\). In interval notation, this is written as \([-2, 3) \cup (3, \infty)\). Here's a breakdown:

• \([-2, 3)\) means all numbers from \(-2\) to \(3\), including \(-2\) but not \(3\).
• \((3, \infty)\) means all numbers greater than \(3\). \(\infty\) is never included in interval notation because it is not an actual number.

Interval notation provides a concise way to convey a lot of information about number sets.

Domain Constraints

Domain constraints are conditions that a function's input values (x-values) must satisfy for the function to be defined and produce real numbers. These constraints stem from mathematical properties, such as the need to avoid taking the square root of a negative number or dividing by zero.In the function \(g(x) = \frac{\sqrt{2+x}}{3-x}\), there are two main domain constraints:

• The square root imposes the condition \(2+x \geq 0\), leading to \(x \geq -2\), ensuring the radicand is non-negative.
• The rational function imposes that the denominator cannot be zero. Thus, \(3-x eq 0\), resulting in \(x eq 3\).

To find the overall domain, you must consider both constraints together, which involves combining the inequalities. The final domain is \([-2, 3) \cup (3, \infty)\). This unified approach helps avoid common mistakes when determining where a function is valid and usable. Understanding domain constraints is one of the foundational skills for mastering calculus and algebra.