Domain and Range of Functions

Introduction

Key Concepts

1. Definition of a Function

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Formally, a function \( f \) from a set \( X \) to a set \( Y \) is written as \( f: X \rightarrow Y \), where \( X \) is the domain and \( Y \) is the codomain.

2. Understanding Domain

The domain of a function is the complete set of possible values of the independent variable, typically denoted as \( x \). It represents all the inputs for which the function is defined. Identifying the domain involves determining all \( x \)-values that do not lead to undefined expressions, such as division by zero or taking the square root of a negative number.

For example, consider the function \( f(x) = \frac{1}{x-2} \). The function is undefined when \( x - 2 = 0 \), i.e., \( x = 2 \). Therefore, the domain is all real numbers except \( x = 2 \), which can be expressed as: $$ \text{Domain: } (-\infty, 2) \cup (2, \infty) $$

3. Understanding Range

The range of a function is the set of all possible output values (dependent variable) that result from using the domain. It encompasses every possible \( y \)-value that the function can produce.

Using the previous example \( f(x) = \frac{1}{x-2} \), as \( x \) approaches 2 from the left, \( f(x) \) approaches negative infinity, and as \( x \) approaches 2 from the right, \( f(x) \) approaches positive infinity. Thus, the range is all real numbers except \( y = 0 \): $$ \text{Range: } (-\infty, 0) \cup (0, \infty) $$

4. Techniques to Determine Domain and Range

• For Polynomial Functions: The domain and range of polynomial functions are typically all real numbers, unless there is a logical restriction.
• For Rational Functions: Identify values that make the denominator zero. These values are excluded from the domain.
• For Radical Functions: Ensure that the expression inside the radical is non-negative (for even roots) or defined according to the context.
• For Trigonometric Functions: Consider the periodic nature and any restrictions based on the function’s definition.

5. Composite Functions

When dealing with composite functions, \( f(g(x)) \), the domain is determined by both the domain of \( g(x) \) and the domains of \( f \) when \( g(x) \) is the input. Ensure that \( g(x) \) is within the domain of \( f \).

6. Inverse Functions

For a function to have an inverse, it must be bijective (both injective and surjective). The domain of the inverse function is the range of the original function, and vice versa.

7. Graphical Interpretation

On a graph, the domain corresponds to the extent of the graph along the \( x \)-axis, while the range corresponds to the extent along the \( y \)-axis. Identifying asymptotes, intercepts, and behavior at infinity helps in determining the domain and range.

8. Practical Examples

• Example 1: Find the domain and range of \( f(x) = \sqrt{5 - x} \).
  • Solution:
    • Inside the square root must be non-negative: \( 5 - x \geq 0 \) ⇒ \( x \leq 5 \).
    • Domain: \( (-\infty, 5] \).
    • Range: \( [0, \infty) \), since the square root function outputs non-negative values.
• Example 2: Determine the domain and range of \( f(x) = \frac{\sqrt{x+3}}{x-1} \).
  • Solution:
    • The numerator requires \( x + 3 \geq 0 \) ⇒ \( x \geq -3 \).
    • The denominator cannot be zero: \( x \neq 1 \).
    • Domain: \( [-3, 1) \cup (1, \infty) \).
    • Range: To find the range, solve for \( y = \frac{\sqrt{x+3}}{x-1} \). This requires more advanced techniques, but generally, the range excludes values that cannot be achieved by the function. Detailed analysis is required for specific ranges.

9. Common Mistakes

• Ignoring restrictions from the denominator in rational functions.
• Overlooking restrictions imposed by even-order roots.
• Assuming that all polynomial functions have all real numbers as their domain and range without verifying.
• Failing to consider the composite nature of functions when determining domains and ranges.

10. Advanced Concepts

In more complex scenarios, such as piecewise functions or functions involving logarithms and exponentials, determining the domain and range may require a deeper understanding of function behavior and transformations. Utilizing calculus concepts like limits can aid in analyzing the boundaries of domains and ranges.

Comparison Table

Summary and Key Takeaways