# How to find the domain and range of a function?

#### Understand the Problem

The question is asking for the methods to determine the domain and range of a mathematical function. The domain refers to the set of all possible input values (x-values) for which the function is defined, while the range refers to the set of all possible output values (y-values) that the function can produce. To solve this, we will typically analyze the function's equation and any restrictions on its variables.

Analyze input restrictions for domain; examine function behavior for range

To find the domain and range of a function, follow these general steps: 1) For the domain, identify any restrictions on the input values, such as values that would make denominators zero or negative numbers under even roots. 2) For the range, analyze the function's behavior, including its graph if possible, to determine the possible output values.

#### Steps to Solve

1. Determine the domain

To find the domain:

2. Look for restrictions on input values (x)

3. Check for:

• Denominators that could be zero
• Even roots of negative numbers
• Logarithms of non-positive numbers
4. Express the domain using interval notation or set-builder notation

Example: For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except 2, as $x \neq 2$.

1. Analyze the range

To find the range:

2. Examine the function's behavior

3. Consider:

• Limitations on output values (y)
• Asymptotes
• Maximum or minimum values
4. Graph the function if helpful

5. Express the range using interval or set-builder notation

Example: For $f(x) = x^2$, the range is $y \geq 0$ or $[0,\infty)$, as squares are always non-negative.

1. Special function types

For certain types of functions, you can use specific methods:

2. Linear functions ($f(x) = mx + b$): Domain is all real numbers, range is all real numbers

3. Quadratic functions ($f(x) = ax^2 + bx + c$): Domain is all real numbers, range depends on the direction of the parabola

4. Rational functions: Check for vertical asymptotes (domain) and horizontal asymptotes (range)

5. Trigonometric functions: Use periodicity and known ranges (e.g., $-1 \leq \sin(x) \leq 1$)

To find the domain and range of a function, follow these general steps: 1) For the domain, identify any restrictions on the input values, such as values that would make denominators zero or negative numbers under even roots. 2) For the range, analyze the function's behavior, including its graph if possible, to determine the possible output values.

Understanding domain and range is crucial for interpreting functions in various contexts, from pure mathematics to applied fields like physics and economics. These concepts help in visualizing function behavior and determining where functions are valid or meaningful in real-world applications.

#### Tips

Common mistakes include:

1. Forgetting to check for restrictions in the domain, especially with complex functions.
2. Misinterpreting the range of a function without considering its entire behavior.
3. Incorrectly applying domain and range concepts to implicit functions or relations that are not functions.
4. Not expressing the domain and range in proper mathematical notation (interval or set-builder notation).
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