Problem 51 - 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 of the function is \([-2, 3) \cup (3, \infty)\).

Step 1: Understanding the function's components

The function given is \( g(x) = \frac{\sqrt{2+x}}{3-x} \). It consists of a square root function in the numerator and a rational expression. To find the domain, we need to ensure that the square root is defined (non-negative) and the denominator is not zero.

Step 2: Finding conditions for the square root

The square root \( \sqrt{2+x} \) is defined when the expression inside is non-negative. This means: \[2 + x \geq 0\]Solving this inequality, we subtract 2 from both sides to find:\[x \geq -2\]

Step 3: Identifying conditions for the denominator

The denominator \( 3-x \) cannot be equal to zero as it would make the fraction undefined. Therefore, we set up the inequality:\[3 - x eq 0\]Solving for \( x \), we find:\[x eq 3\]

Step 4: Combining the conditions

The domain is where both conditions for the numerator and denominator are satisfied. We need \( x \geq -2 \) and \( x eq 3 \). Combining these, the domain of \( g(x) \) in interval notation is:\[[-2, 3) \cup (3, \infty)\]

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In the exercise, our function is an example of a rational expression because it involves a fraction with a square root in the numerator and a linear expression in the denominator. It's important to remember that rational expressions can sometimes be undefined. This usually occurs when the denominator equals zero.
For instance, in the given function, the denominator is \(3-x\). This means that the function is undefined if \(3-x=0\), which would happen when \(x=3\). Therefore, to ensure the function is defined, \(x\) must not equal 3. This is a crucial step in finding the domain of a rational function.
In summary, whenever you're dealing with rational expressions, always check that the denominator isn't zero to avoid undefined scenarios.

Square Root Functions

Square root functions include expressions like \(\sqrt{2+x}\) seen in the function \(g(x)\). The main condition for a square root function to be valid is that whatever is under the radical must be non-negative. This is because the square root of a negative number is not a real number in the context of basic algebra.
For the exercise function, \(\sqrt{2+x}\), we need the expression \(2+x\) to be greater than or equal to zero for the square root to be valid. Solving \(2+x \geq 0\) gives us \(x \geq -2\). This means the smallest \(x\) can be is -2, and it can go as large as necessary, provided the original rational condition is also met.
Square root functions are critical in finding domains, as you'll often need to both avoid negative values under the radical and ensure other conditions from the function's structure, like keeping the denominator non-zero, are also satisfied.

Interval Notation

Interval notation is a mathematical shorthand that represents a range of values for which a function is defined. It’s used to conveniently specify the domain of functions. In the original exercise, interval notation is used to express the domain of the function \(g(x) = \frac{\sqrt{2+x}}{3-x}\).
After determining that \(x \geq -2\) from the square root condition and \(x eq 3\) from the rational expression condition, we combine these constraints into interval notation. The function is defined wherever these conditions hold true simultaneously.
The domain is written as \([-2, 3) \cup (3, \infty)\). This means that \(x\) can be from -2 up to, but not including, 3, and then any number larger than 3. The parentheses \((\) and \()\) signify that the endpoint is not included, while brackets \([\) and \(]\) indicate inclusion. For example, the point -2 is included, but 3 is not. Using interval notation helps students quickly understand and communicate the precise limits of a function's domain.