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# Domain range

In mathematics, the domain of a function is the complete set of possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) the function can produce. To determine the domain, look for values that might cause division by zero or negative square roots, and for the range, assess the outputs based on the functional formula. Understanding both concepts is essential for analysing the behaviour of functions and solving equations effectively.

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## Domain range: Definition

Understanding the domain and range of a function is crucial when studying mathematics. Both concepts are foundational elements of functions, which are essential in various mathematical applications.

### Domain

The domain of a function refers to the set of all possible input values (usually denoted as x) that the function can accept without causing undefined or non-real values. Essentially, it is the complete set of values for which the function is defined.For example, consider the function:

• $$f(x) = \sqrt{x}$$
For this function, x must be greater than or equal to 0, because the square root of a negative number is not a real number. Therefore, the domain of $$f(x) = \sqrt{x}$$ is:Domain: $$x \geq 0$$

Sometimes the domain is explicitly given, but other times you may need to determine it yourself by identifying values that make the function undefined.

### Range

The range of a function, on the other hand, is the set of all possible output values (usually denoted as y) that the function can produce. It is determined by the values that come out of the function when the entire domain is considered.Sticking with our last example:

• $$f(x) = \sqrt{x}$$
The output values (or the range) for this function are all non-negative real numbers, as the square root function produces results that are zero or positive.Range: $$y \geq 0$$

Domain: The set of all possible input values for which a function is defined. Range: The set of all possible output values produced by a function.

Let's consider another example with a different function:For a quadratic function such as:

• $$f(x) = x^2 - 4$$
The domain of this function is all real numbers because you can input any real number into $$x$$ and get a defined output. However, the range of this function is limited to values that $$x^2 - 4$$ can produce. Since $$x^2$$ is always non-negative (zero or positive), the smallest value $$f(x)$$ can take is when $$x = 0$$, which gives us $$f(x) = -4$$. Thus, the range of $$f(x) = x^2 - 4$$ is all real numbers greater than or equal to -4.Domain: All real numbers Range: $$y \geq -4$$

## What is domain and range in maths?

Understanding the domain and range of a function is vital in mathematics. Both concepts form the foundational elements of functions, which are key in various mathematical applications.Each function has a specific domain and range, determining the permissible inputs and possible outputs, respectively.

### Domain

The domain of a function comprises all possible input values (usually denoted as x) that the function can accept without causing undefined or non-real values. Essentially, it is the complete set of values for which the function is defined.

Consider the function:

• $$f(x) = \sqrt{x}$$
For this function, x must be greater than or equal to 0 because the square root of a negative number is not a real number. Therefore, the domain of $$f(x) = \sqrt{x}$$ is:Domain: $$x \geq 0$$

Remember, sometimes the domain is explicitly given, but other times you may need to determine it yourself by identifying values that make the function undefined.

### Range

The range of a function refers to the set of all possible output values (usually denoted as y) that the function can produce. It is determined by the values that come out of the function when the entire domain is considered.

Consider the function again:

• $$f(x) = \sqrt{x}$$
The output values (or the range) for this function are all non-negative real numbers, as the square root function produces results that are zero or positive.Range: $$y \geq 0$$

Domain: The set of all possible input values for which a function is defined. Range: The set of all possible output values produced by a function.

Let's consider a more complex example with a different function:For a quadratic function such as:

• $$f(x) = x^2 - 4$$
The domain of this function is all real numbers because you can input any real number into $$x$$ and get a defined output. However, the range of this function is limited to values that $$x^2 - 4$$ can produce. Since $$x^2$$ is always non-negative (zero or positive), the smallest value $$f(x)$$ can take is when $$x = 0$$, which gives us $$f(x) = -4$$. Thus, the range of $$f(x) = x^2 - 4$$ is all real numbers greater than or equal to -4.Domain: All real numbers Range: $$y \geq -4$$

## How to find domain and range of a function

Finding the domain and range of a function is a critical skill that helps you understand the behaviour of different types of functions. This knowledge is applicable in various fields such as engineering, science, and more.

### Identifying the Domain

To find the domain of a function, follow these steps:

• Identify the type of function (e.g., polynomial, rational, square root).
• Determine which values of x would make the function undefined (such as division by zero or taking the square root of a negative number).
• Exclude any values that make the function undefined.
Let's consider some examples to clarify these steps.

Example 1:

• Function: $$f(x) = \frac{1}{x - 2}$$
For this function, the denominator becomes zero when x = 2, which makes the function undefined. Thus, the domain excludes 2.Domain: All real numbers except 2.Example 2:
• Function: $$f(x) = \sqrt{x + 3}$$
Here, the expression inside the square root must be non-negative, so x + 3 must be greater than or equal to 0.Solve: $$x + 3 \geq 0$$Domain: $$x \geq -3$$

Always check for both explicit and implicit constraints that affect the domain of the function.

### Identifying the Range

Finding the range of a function can be more challenging than finding the domain. The general process involves considering the function's behaviour over its entire domain and determining the possible output values y. Follow these steps:

• Identify the domain of the function.
• Determine how the function behaves as x approaches different values within the domain.
• Find the minimum and maximum output values if they exist.
Let's look at some examples to illustrate this.

Example 1:

• Function: $$f(x) = x^2$$
The function $$f(x) = x^2$$ produces output values that are always non-negative (since squaring any real number results in a positive or zero value).Range: $$y \geq 0$$Example 2:
• Function: $$f(x) = \frac{1}{x}$$
In this case, as $$x$$ approaches zero from either direction, the function values become very large either positively or negatively, but $$f(x)$$ does not ever reach zero.Range: All real numbers except 0.

Some special functions might require more complex techniques to find their range, such as inverse functions or completing the square for quadratic functions.

• For inverse functions, finding the range of the original function can involve reflecting the graph over the line $$y = x$$.
• In quadratics, sometimes completing the square helps to identify the vertex, which indicates the minimum or maximum value, aiding in determining the range.
For instance:Quadratic function: $$f(x) = x^2 - 4x + 3$$First, convert it to vertex form by completing the square:$$f(x) = (x - 2)^2 - 1$$Now, the vertex is at (2, -1), meaning minimum value of $$f(x)$$ is -1. Therefore, the range is all real numbers greater than or equal to -1.

## Domain and range examples

Understanding the domain and range of a function is crucial when studying mathematics. Both concepts form foundational elements of functions, widely used in various mathematical applications.

### What is domain and range?

The domain of a function consists of all possible input values (usually denoted as x) that the function can accept without causing undefined or non-real values. The range of a function, on the other hand, includes all possible output values (usually denoted as y) that the function can produce.

### Definition of domain and range

Domain: The set of all possible input values for which a function is defined. Range: The set of all possible output values produced by a function.

### Steps to find domain and range

Identifying the domain and range of a function can be achieved through systematic steps.To find the domain, generally follow these steps:

• Identify the function type (e.g., polynomial, rational, square root).
• Determine which values of x would make the function undefined (such as division by zero or taking the square root of a negative number).
• Exclude any values that make the function undefined.
For finding the range, consider these points:
• Identify the domain of the function.
• Determine how the function behaves as x approaches different values within the domain.
• Find the minimum and maximum output values if they exist.
Let's explore some practical examples.

### Practical examples of domain and range

Example 1:Function: $$f(x) = \frac{1}{x - 2}$$For this function, the denominator becomes zero when x = 2, making the function undefined.Domain: All real numbers except 2.To find the range:

• The function can produce any output except zero, given that dividing by x - 2 cannot result in zero.
Range: All real numbers except 0.

Always check for both explicit and implicit constraints that affect the domain of the function.

Let's consider a more complex example:Quadratic function: $$f(x) = x^2 - 4x + 3$$First, convert to vertex form by completing the square:$$f(x) = (x - 2)^2 - 1$$Now, the vertex is at (2, -1), indicating that the minimum value of $$f(x)$$ is -1.Domain: All real numbersRange: $$y \geq -1$$For rational functions like $$f(x) = \frac{1}{x}$$, the domain excludes x = 0 because dividing by zero is undefined, while the range includes all real numbers except zero.

## Domain range - Key takeaways

• Domain range: Refers to the set of all possible input values (domain) and output values (range) of a function.
• Domain and range of a function: The domain includes all possible input values (x) that make the function defined, while the range includes all output values (y) that the function can produce.
• How to find domain and range: Identify the type of function, determine which values make the function undefined, and exclude them to find the domain; to find the range, consider the function's behaviour over its domain.
• Examples: For the function f(x) = sqrt(x), the domain is x ≥ 0 and the range is y ≥ 0; for f(x) = x^2 - 4, the domain is all real numbers and the range is y ≥ -4.
• Definition of domain and range: The domain is the set of all possible input values for which a function is defined, and the range is the set of all possible output values produced by a function.

#### Flashcards in Domain range 12

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What is the difference between domain and range in a function?
The domain of a function consists of all the input values for which the function is defined, while the range consists of all the output values the function can produce. Essentially, the domain is the set of possible x-values, and the range is the set of resulting y-values.
How do you determine the domain and range of a function?
To determine the domain, identify all possible input values for which the function is defined. For the range, find all possible output values the function can produce. Analyse the function's expression or graph, considering restrictions such as division by zero or negative square roots.
What are the domain and range of a quadratic function?
The domain of a quadratic function is all real numbers, denoted as \$$(-\\infty, \\infty) \$$. The range depends on the direction of the parabola: \$$[k, \\infty) \$$ if it opens upwards, and \$$(-\\infty, k] \$$ if it opens downwards, where \$$k \$$ is the vertex's y-coordinate.
Can the domain and range of a function be the same?
Yes, the domain and range of a function can be the same. This occurs in functions like the identity function f(x) = x, where each input value maps to itself.
Can the domain of a function include complex numbers?
Yes, the domain of a function can include complex numbers. Functions of complex variables are studied in complex analysis, where the domain can be all complex numbers or a subset of them.

## Test your knowledge with multiple choice flashcards

How can you find the vertex of a quadratic function to determine its range?

For the function $$f(x) = \sqrt{x}$$, what is the domain?

What is the range of the function $$f(x) = x^2 - 4$$?

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