Domain and Range of Functions

Introduction

Key Concepts

1. Definitions

In mathematics, a function is a relation that uniquely associates elements of one set with elements of another set. The domain of a function is the set of all possible input values (typically represented by 'x') for which the function is defined. In other words, it encompasses all the permissible values that 'x' can take without causing any mathematical inconsistencies, such as division by zero or taking the square root of a negative number.

The range of a function, on the other hand, is the set of all possible output values (typically represented by 'f(x)' or 'y') that result from substituting the domain values into the function. It represents the potential outcomes or results produced by the function.

2. Finding the Domain

Determining the domain of a function involves identifying all real numbers 'x' for which the function is defined. This process varies depending on the type of function:

• Polynomial Functions: Polynomials, such as $f(x) = x^2 + 3x + 2$, have domains of all real numbers since they are defined for every real input.
• Rational Functions: For functions like $f(x) = \frac{1}{x-2}$, the domain excludes values that cause the denominator to be zero. Here, $x \neq 2$.
• Radical Functions: Functions involving square roots, like $f(x) = \sqrt{x-1}$, require the expression inside the radical to be non-negative. Thus, the domain is $x \geq 1$.
• Logarithmic Functions: For $f(x) = \log(x)$, the argument must be positive, so the domain is $x > 0$.

3. Finding the Range

Determining the range requires analyzing the possible output values based on the domain and the behavior of the function:

• Polynomial Functions: The range of an even-degree polynomial like $f(x) = x^2$ is $y \geq 0$, while an odd-degree polynomial covers all real numbers.
• Rational Functions: For $f(x) = \frac{1}{x}$, as $x$ approaches zero from the positive side, $y$ approaches infinity, and from the negative side, $y$ approaches negative infinity. Thus, the range is all real numbers except $y = 0$.
• Radical Functions: For $f(x) = \sqrt{x-1}$, since the output is always non-negative, the range is $y \geq 0$.
• Logarithmic Functions: The range of $f(x) = \log(x)$ is all real numbers, as the logarithm can take any real value.

4. Techniques for Determining Domain and Range

4.1. Using Graphs

Graphical analysis is a powerful tool for identifying the domain and range of functions:

• Domain: Look for the set of all possible 'x' values where the function has points on the graph.
• Range: Observe the set of all possible 'y' values that the function attains.

For example, the graph of $f(x) = \sqrt{x-1}$ starts at $x = 1$ and extends to the right indefinitely, indicating a domain of $x \geq 1$ and a range of $y \geq 0$.

4.2. Algebraic Methods

Algebraic techniques involve manipulating the function to solve for restrictions on 'x' or 'y':

• Domain Restrictions: Set the denominator not equal to zero, ensure arguments of even roots are non-negative, and arguments of logarithms are positive.
• Range Determination: Solve the function equation $y = f(x)$ for 'x' in terms of 'y' and identify the possible 'y' values.

Consider $f(x) = \frac{\sqrt{x}}{x-1}$. To find the domain:

• The square root requires $x \geq 0$.
• The denominator demands $x \neq 1$.

Therefore, the domain is $x \geq 0$ and $x \neq 1$. To find the range, set $y = \frac{\sqrt{x}}{x-1}$ and solve for 'x': $$ y(x - 1) = \sqrt{x} $$ This equation may require further analysis or graphical methods to determine the range.

5. Common Mistakes and How to Avoid Them

When determining domain and range, several common errors can arise:

• Ignoring Denominator Restrictions: Failing to exclude values that make denominators zero can lead to incorrect domain identification.
• Misinterpreting Radical Functions: Not setting the expression under the square root to be non-negative can result in wrong domain and range.
• Overlooking Logarithmic Constraints: Forgetting that the argument of a logarithm must be positive can cause errors in finding the domain.
• Incorrect Range Analysis: Assuming the range is all real numbers without considering the function's behavior can lead to mistakes.

To avoid these mistakes, systematically analyze each component of the function, use graphical representations, and cross-verify results with multiple methods.

6. Examples

6.1. Example 1: Finding Domain and Range

Consider the function $f(x) = \frac{2x + 3}{x^2 - 4}$.

Finding the Domain:

• Set the denominator not equal to zero: $x^2 - 4 \neq 0$.
• Solve: $x \neq \pm 2$.
• Domain: $x \in \mathbb{R}, x \neq -2, 2$.

Finding the Range:

To find the range, solve for 'x' in terms of 'y': $$ y = \frac{2x + 3}{x^2 - 4} $$ $$ y(x^2 - 4) = 2x + 3 $$ $$ yx^2 - 4y - 2x - 3 = 0 $$ This is a quadratic in 'x'. For real solutions to exist, the discriminant must be non-negative: $$ \Delta = (-2)^2 - 4(y)(-4y - 3) \geq 0 $$ $$ 4 + 16y^2 + 12y \geq 0 $$ $$ 16y^2 + 12y + 4 \geq 0 $$ Since the discriminant here is always positive, there are real solutions for all real 'y'. Therefore, the range is all real numbers.

6.2. Example 2: Finding Domain and Range

Consider the function $f(x) = \sqrt{5 - x} + \ln(x)$.

Finding the Domain:

• The square root requires $5 - x \geq 0 \Rightarrow x \leq 5$.
• The natural logarithm requires $x > 0$.
• Combine the two conditions: $0 < x \leq 5$.
• Domain: $0 < x \leq 5$.

Finding the Range:

Analyzing the range involves understanding the behavior of both $\sqrt{5 - x}$ and $\ln(x)$ within the domain $0 < x \leq 5$:

• As $x$ approaches $0^+$, $\sqrt{5 - x} \to \sqrt{5}$ and $\ln(x) \to -\infty$, so $f(x) \to -\infty$.
• At $x = 5$, $\sqrt{5 - 5} = 0$ and $\ln(5) \approx 1.609$, so $f(5) \approx 1.609$.

Since $f(x)$ is continuous on $(0, 5]$ and approaches $-\infty$ as $x$ approaches $0^+$ while attaining a finite value at $x = 5$, the range of $f(x)$ is $(-\infty, \ln(5)]$.

Comparison Table

Summary and Key Takeaways