Question

Find the domain of the function. \(g(x)=\frac{1}{x}-\frac{3}{x+2}\)

Short answer

The domain of the function \(g(x) = \frac{1}{x} - \frac{3}{x+2}\) is all real numbers except \(x = 0\) and \(x = -2\). In interval notation, the domain is \((- \infty, -2) \cup (-2, 0) \cup (0, +\infty)\).

Step-by-step solution

Identify the Restrictions in the Denominator

The denominators of the fractions in the function are 'x' and 'x + 2'. A fraction is undefined when the denominator is zero. So, to find where the function is undefined, set each denominator equal to zero and solve for 'x'. This will give us 'x = 0' from '\(\frac{1}{x}\)' and 'x = -2' from '\(\frac{3}{x + 2}\)'.

Test the Restrictions

Apply the values 'x = 0' and 'x = -2' to the function, which yield undefined results. Therefore, these values are not in the domain of the function.

Write the Domain

The domain of the function includes all real numbers except the restrictions that we found out in Step 1 & 2. Therefore, the domain of \(g(x) = \frac{1}{x} - \frac{3}{x+2}\) is \(x \neq 0\) and \(x \neq -2\). In interval notation, the domain is \((- \infty, -2) \cup (-2, 0) \cup (0, +\infty)\).

Key concepts

Function Domain Restrictions

The domain of a function represents the set of all possible input values that the function can accept without leading to undefined or nonsensical answers. Function domain restrictions occur due to specific values that force the function into an undefined state, such as division by zero, or taking the square root of a negative number in the realm of real numbers.

In the case of the function g(x) = \( \frac{1}{x} - \frac{3}{x+2} \), the restrictions come into play as a result of the denominators in the fractions. A non-zero denominator is a fundamental rule in mathematics because dividing by zero has no defined value. Therefore, to find the domain, we first determine the values for x that make any denominator zero, which in this function are x = 0 and x = -2.

Understanding and identifying domain restrictions is crucial for correctly graphing functions and accurately performing calculus operations such as finding limits and derivatives. Knowing these constraints avoids errors in computations and helps better grasp the behavior of functions across different intervals.

Undefined Function Values

A function value becomes undefined when an operation within the function does not yield a particular result. This undefined result disrupts the normal continuity of the function's graph and is not included in the domain.

In algebraic functions like g(x) from the example, undefined values typically arise from zero denominators or negative radicands in even roots. For instance, g(x) = \( \frac{1}{x} - \frac{3}{x+2} \) becomes undefined for x = 0 and x = -2 because those inputs create zero denominators.

Becoming familiar with operations that can yield undefined values is essential for students to master the concept of domains. This empowers them to predict the behavior of functions before even graphing them and understand more complex mathematical ideas involved in higher levels of math, such as limits and continuity in calculus.

Interval Notation

Interval notation is a concise way to communicate the set of all real numbers that fall within a specific range, including or excluding certain points. It streamlines the expression of domains, ranges, and other sets within mathematics.

For the function g(x), the domain using interval notation is expressed as (-∞, -2) ∪ (-2, 0) ∪ (0, +∞). Here's a breakdown of what this means:

• The parentheses () indicate that the endpoints are not included in the set because the function is undefined at x = 0 and x = -2.
• The union symbol ∪ communicates that the domain is the combination of multiple intervals where the function is defined.
• The infinity symbols -∞ and +∞ show that there is no upper or lower bound on the values of x within the defined intervals.

Mastering interval notation is an important skill for students, as it is widely used in calculus and higher mathematics to simplify the communication of complex sets and ensure precision in describing the scope of functions and equations.