domain and range of graphs worksheet

2 min read 24-11-2024

Understanding domain and range is crucial for grasping fundamental concepts in algebra and beyond. This worksheet will guide you through various examples, helping you master identifying the domain and range of different types of graphs. We'll cover linear functions, quadratic functions, absolute value functions, and more. Let's dive in!

What are Domain and Range?

Before we tackle the worksheet, let's clarify the definitions:

Domain: The set of all possible input values (x-values) for a function. Think of it as the function's allowed "x-territory."
Range: The set of all possible output values (y-values) for a function. This represents the function's resulting "y-territory."

Types of Graphs and Finding Domain & Range

Let's explore how to find the domain and range for different types of graphs. Each example will demonstrate a clear method to tackle these problems.

1. Linear Functions

Example: Consider the linear function y = 2x + 1.

Domain: Linear functions typically have a domain of all real numbers (-∞, ∞). There are no restrictions on the x-values you can input.
Range: Similarly, the range of this linear function is all real numbers (-∞, ∞). The line extends infinitely in both the positive and negative y-directions.

Worksheet Practice: Find the domain and range of the following linear functions:

y = x - 5
y = -3x + 2
y = 1/2x

2. Quadratic Functions

Example: Consider the quadratic function y = x²

Domain: The domain is again all real numbers (-∞, ∞). You can square any real number.
Range: The range, however, is restricted. Since x² is always non-negative, the range is [0, ∞). The parabola opens upwards, starting at y = 0 and extending infinitely upwards.

Worksheet Practice: Find the domain and range of the following quadratic functions:

y = x² - 4
y = -x² + 2
y = 2(x + 1)² - 3

3. Absolute Value Functions

Example: Consider the absolute value function y = |x|

Domain: The domain is all real numbers (-∞, ∞). You can find the absolute value of any real number.
Range: The range is [0, ∞). The absolute value is always non-negative.

Worksheet Practice: Find the domain and range of the following absolute value functions:

y = |x| + 1
y = -|x| + 2
y = 2|x - 1|

4. Functions with Restrictions

Sometimes, functions have explicit restrictions on their domain.

Example: Consider the function y = 1/x

Domain: The domain is all real numbers except x = 0. You cannot divide by zero. We represent this as (-∞, 0) U (0, ∞).
Range: Similarly, the range is all real numbers except y = 0. The function never equals zero. This is represented as (-∞, 0) U (0, ∞).

Worksheet Practice: Find the domain and range of the following functions:

y = 1/(x-2)
y = √x (Note: you cannot take the square root of a negative number)
y = 1/√x

5. Graphs with Discrete Points

Example: Consider a graph consisting of only a few points: {(1,2), (3,4), (5,6)}

Domain: The domain is the set of x-values: {1, 3, 5}
Range: The range is the set of y-values: {2, 4, 6}

Worksheet Practice: Find the domain and range of the following sets of points:

{(2,1), (4,3), (6,5)}
{(-1,0), (0,1), (1,0)}
{(0,0), (1,1), (2,4), (3,9)}

Interval Notation

Remember to express your answers using correct interval notation.

(a, b): Open interval, excluding a and b.
[a, b]: Closed interval, including a and b.
(a, b]: Half-open interval, including b, but excluding a.
[a, b): Half-open interval, including a, but excluding b.
(-∞, a): All numbers less than a.
(a, ∞): All numbers greater than a.

This worksheet provides a solid foundation for understanding domain and range. Remember to practice regularly to solidify your skills. Good luck!