domain and range answer key

2 min read 04-02-2025

Understanding domain and range is fundamental to grasping the core concepts of functions in mathematics. This comprehensive guide serves as an answer key, explaining these concepts clearly and providing examples to solidify your understanding. We'll explore various function types and how to determine their domain and range effectively.

What are Domain and Range?

Before diving into examples, let's define our key terms:

Domain: The domain of a function is the set of all possible input values (often denoted by 'x') for which the function is defined. Think of it as the acceptable inputs the function can handle without producing an error (like division by zero or taking the square root of a negative number).
Range: The range of a function is the set of all possible output values (often denoted by 'y' or 'f(x)') that the function can produce. It's the collection of all possible results you get after plugging in values from the domain.

Determining Domain and Range: A Step-by-Step Approach

The method for finding the domain and range depends on the type of function. Here's a breakdown of common scenarios:

1. Polynomial Functions (e.g., f(x) = x² + 2x + 1)

Domain: Polynomial functions are defined for all real numbers. There are no restrictions on the input values. Therefore, the domain is all real numbers, often written as (-∞, ∞) in interval notation or ℝ.
Range: The range depends on the specific polynomial. For quadratic functions like the example above, the range is often restricted. Analyzing the graph or completing the square can help determine the minimum or maximum value, defining the range's boundaries. For the example, the parabola opens upwards, thus the range is [minimum value, ∞).

2. Rational Functions (e.g., f(x) = 1/(x-2))

Domain: The key here is to identify values that would make the denominator zero. In the example, x = 2 would cause division by zero, making the function undefined. Therefore, the domain is all real numbers except 2, written as (-∞, 2) U (2, ∞).
Range: Rational functions often have horizontal and vertical asymptotes, influencing their range. Analyzing the function's behavior around asymptotes and potential extrema will determine the range. In this case, the range is (-∞, 0) U (0, ∞).

3. Radical Functions (e.g., f(x) = √(x+3))

Domain: For even-indexed roots (square roots, fourth roots, etc.), the radicand (the expression inside the root) must be non-negative. So, for our example, x + 3 ≥ 0, which means x ≥ -3. The domain is [-3, ∞).
Range: The principal square root is always non-negative. Therefore, the range of f(x) = √(x+3) is [0, ∞).

4. Trigonometric Functions

The domain and range of trigonometric functions (sine, cosine, tangent, etc.) are specific and often cyclic. Consult a trigonometric functions reference for detailed explanations. For example:

sin(x): Domain: (-∞, ∞); Range: [-1, 1]
cos(x): Domain: (-∞, ∞); Range: [-1, 1]
tan(x): Domain: all real numbers except odd multiples of π/2; Range: (-∞, ∞)

Practice Problems

To test your understanding, try determining the domain and range of these functions:

f(x) = x³ - 5x + 2
g(x) = 1/(x² - 4)
h(x) = √(4 - x)
k(x) = 2sin(x)

Conclusion

Mastering the concepts of domain and range is crucial for understanding and working with functions in mathematics. By systematically analyzing the function type and potential restrictions, you can confidently determine the acceptable inputs and corresponding outputs. Remember to always consider potential issues like division by zero and negative radicands under even roots. Practice is key! Work through various examples and gradually increase complexity to build your proficiency.