Domain and Range of Functions

What are Domain and Range?

Let's begin by solidifying our definitions. In the simplest terms, for a function $f(x)=y$:

• The domain is the set of all possible input values ($x$-values).
• The range is the set of all possible output values ($y$-values).

Your background in IT provides a useful parallel. When you design a system or write a program, you define the types of data an input field will accept (e.g., an integer, a string, a value within a certain range). This is "input validation," and it prevents the program from crashing. A function's domain serves the same purpose: it's the set of all inputs for which the function is mathematically well-defined and produces a real number.

To formalize this, please review the definitions from the LibreTexts resource.

The image below provides a fantastic visual summary of the domain and range for several common "parent" functions. Take a moment to study it and look for patterns. For instance, notice how functions like the square root and reciprocal have restrictions on their domains, while others like the cubic function do not.

Finding Domain and Range from a Graph

The graph of a function gives us a direct picture of its behavior. We can determine the domain and range by seeing how far the function "stretches" along the x and y axes.

• To find the domain: Imagine squashing the entire graph vertically onto the x-axis. The portion of the x-axis that the graph covers is the domain.
• To find the range: Imagine squashing the entire graph horizontally onto the y-axis. The portion of the y-axis that the graph covers is the range.

The following image illustrates this projection method for a cubic function and a square root function.

This graphical method is powerful, especially for complex graphs with breaks or endpoints. The following video provides a concise tutorial on applying this technique in various scenarios. Pay close attention to how open circles, closed circles, and arrows are handled using interval notation.

Finding Domain from an Algebraic Formula

While graphs are intuitive, you will often need to determine the domain directly from a function's algebraic formula. The strategy is to assume the domain is all real numbers, $(-\infty ,\infty )$, and then look for values of $x$ that would cause a mathematical problem. For the functions we'll study in this course, there are two primary rules to enforce:

1. You cannot divide by zero. For any rational function (a fraction), the denominator can never be equal to 0.
2. You cannot take an even-indexed root of a negative number. For functions involving square roots, fourth roots, etc., the expression inside the radical (the radicand) must be greater than or equal to 0.

Let's see how these rules are applied. The following comprehensive video covers many function types, but we'll focus on the ones most relevant to these two rules.

Your textbook reinforces these exact points. You can read through the following short example from the Stewart text to see the same logic presented in a textbook format.

What about other functions?

• Polynomials (e.g., $f(x)=3x+2$ or $g(x)=x^2+2x-8$): Since there is no division by a variable and no square roots, there are no restrictions. The domain of any polynomial function is all real numbers, $(-\infty ,\infty )$.
• Odd-indexed roots (e.g., $f(x)=\sqrt[3]{x}$): You can take the cube root of a negative number (e.g., $\sqrt[3]{-8}=-2$). Therefore, odd-indexed roots do not impose any restrictions on the domain.

A Note on Finding the Range Algebraically

Finding the range from a formula can be more challenging than finding the domain. There isn't a single, universal method. However, two common strategies are:

1. Use knowledge of parent functions and transformations. If you know the range of $y=x^2$ is $[0,\infty )$, then the range of $y=x^2+3$ is $[3,\infty )$, because the entire graph is shifted up by 3 units. Similarly, since the range of $y=\sqrt{x}$ is $[0,\infty )$, the range of $y=-\sqrt{x}$ is $(-\infty ,0]$ because the graph is reflected across the x-axis.

2. Analyze the function's characteristics. The vertex of a parabola determines its minimum or maximum value, which in turn defines the boundary of its range. A horizontal asymptote of a rational or exponential function often defines a boundary for its range.

The video from The Organic Chemistry Tutor demonstrates this well.

Conclusion

In this lesson, you developed a systematic toolkit for determining the domain and range of functions, a fundamental skill in calculus. You learned to "read" the domain and range from a graph and how to deduce them from an algebraic formula by looking for mathematical restrictions.

Key Takeaways:

• The domain is the set of all valid $x$-values (inputs), while the range is the set of all resulting $y$-values (outputs).
• From a graph, the domain is the projection onto the x-axis, and the range is the projection onto the y-axis. Use interval notation with ( ) for excluded points and [ ] for included points.
• From a formula, the domain is typically found by excluding values that cause division by zero or even roots of negative numbers.
• The range is often determined by analyzing transformations of parent functions or by finding key features like vertices and asymptotes.

In our next lesson, we will continue our investigation of function properties by exploring symmetry. We will learn to identify even, odd, and periodic functions from their graphs and algebraic definitions, adding another layer to our ability to describe and classify functions.