Finding the domain and range of a function is a fundamental skill in mathematics. The domain represents the set of all possible input values (x-values), while the range represents the set of all possible output values (y-values) of a function.

## How to Find the Domain and Range of a Function

In this guide, we will provide easy-to-understand, step-by-step instructions on how to determine the domain and range of various types of functions.

### Step 1: Identify the Type of Function

First, it’s essential to recognize the type of function you’re dealing with, as different types of functions have specific rules for determining their domain and range. Some common function types include:

- Linear functions
- Quadratic functions
- Rational functions
- Exponential functions
- Logarithmic functions
- Trigonometric functions

### Step 2: Determine the Domain

For each type of function, follow these guidelines to find the domain:

#### Linear Functions

The domain of a linear function is always all real numbers, as there are no restrictions on the input values.

#### Quadratic Functions

The domain of a quadratic function is also all real numbers, as there are no restrictions on the input values.

#### Rational Functions

For a rational function, the domain is all real numbers except those that make the denominator equal to zero. To find these values, set the denominator equal to zero and solve for x.

#### Exponential Functions

The domain of an exponential function is all real numbers, as exponential functions are defined for all input values.

#### Logarithmic Functions

For a logarithmic function, the domain is all positive real numbers, since logarithms are only defined for positive input values.

#### Trigonometric Functions

The domain of trigonometric functions depends on the specific function. For example, sine and cosine functions have a domain of all real numbers, while the tangent function has a domain of all real numbers except odd multiples of π/2.

### Step 3: Determine the Range

For each type of function, follow these guidelines to find the range:

#### Linear Functions

The range of a linear function is all real numbers, as there are no restrictions on the output values.

#### Quadratic Functions

The range of a quadratic function depends on the function’s vertex and whether it opens upward or downward. If the parabola opens upward, the range is all real numbers greater than or equal to the y-coordinate of the vertex. If the parabola opens downward, the range is all real numbers less than or equal to the y-coordinate of the vertex.

#### Rational Functions

The range of a rational function is typically all real numbers except for a horizontal asymptote, which is a value that the function never reaches. To find this value, analyze the degree of the numerator and denominator.

#### Exponential Functions

The range of an exponential function is all positive real numbers since exponential functions always produce positive output values.

#### Logarithmic Functions

The range of a logarithmic function is all real numbers, as logarithmic functions can produce any real number as an output value.

#### Trigonometric Functions

The range of trigonometric functions depends on the specific function. For example, the range of sine and cosine functions is [-1, 1], while the range of the tangent function is all real numbers.

## Conclusion

In summary, to find the domain and range of a function, you need to first identify the type of function and then follow the specific guidelines for that function type. This process will help you determine the set of all possible input and output values for any given function, which is an essential skill in mathematics.

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