Understanding Domain and Range of Functions

1. Division by Zero

In mathematics, division by zero is undefined. Therefore, any value of $x$ that causes the denominator of a fraction to become zero must be excluded from the domain.

Example: Consider the function: $f(x) = \frac{5}{x - 4}$

To find the domain, we set the denominator to zero to find the "forbidden" value: $x - 4 = 0$ $x = 4$

Since $x = 4$ would result in division by zero, the domain is all real numbers except $x = 4$. In interval notation, we write this as: $(-\infty, 4) \cup (4, \infty)$

2. Even Roots of Negative Numbers

The square root (or any even root, like a fourth root) of a negative number is not a real number. Therefore, the expression inside an even radical must be greater than or equal to zero.

Example: Consider the function: $g(x) = \sqrt{x + 2}$

To find the domain, we must ensure the radicand (the part inside the root) is non-negative: $x + 2 \geq 0$ $x \geq -2$

The domain is all real numbers greater than or equal to $-2$. In interval notation, this is: $[-2, \infty)$

Determining the Range

Finding the range is often more challenging than finding the domain because it requires you to understand the behavior of the function as a whole. While the domain is about what is allowed, the range is about what is achievable.

There are several strategies to find the range:

1. Analyzing the Function Type

Certain parent functions have inherent ranges. For example:

• Quadratic Functions: $f(x) = x^2$ will never produce a negative number. Therefore, its range is $[0, \infty)$.
• Absolute Value Functions: $f(x) = |x|$ also only produces non-negative results, so its range is $[0, \infty)$.
• Sine and Cosine: These functions oscillate between $-1$ and $1$, meaning their range is restricted to $[-1, 1]$.

2. Using the Graph

If you have a visual representation of the function, finding the range is a matter of looking at the $y$-axis. Scan the graph from the bottom to the top. The lowest point reached by the graph is your minimum $y$-value, and the highest point is your maximum $y$-value.

3. Algebraic Inversion (Advanced)

For more complex rational functions, a common technique is to solve the equation for $x$ in terms of $y$. Once you have $x = \text{some expression of } y$, you can apply the same "division by zero" and "even root" rules to $y$ to find the range.

Summary Table of Restrictions

To help you prepare for exams, use this quick reference guide when analyzing functions:

Exam Tip: When writing your final answer, pay close attention to the difference between parentheses $(,)$ and brackets $[,]$. Use brackets if the number is included (e.g., $\geq$ or $\leq$) and parentheses if the number is excluded (e.g., $>$ or $<$ or when dealing with infinity).