Domain and Range

• Introduction
• What are Domain and Range?
• Domain and Range of a Function
• `3` Ways to Write Domain and Range
• Understanding the Domain of a Function
• How to Find the Domain of a Function?
• Understanding the Range of a Function
• How to Find the Range of a Function?
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

In math, we use the terms domain and range to talk about the possible inputs and outputs of a function. Think of the domain as all the numbers you can plug into a function, and the range as all the numbers you can get out of it. These concepts are super important in math and have lots of cool applications. Let's take a closer look!

What are Domain and Range?

Domain is all about the input values of a function. For a function, it's the set of all possible `x`-values you can plug in and get a valid output. You can see it as the set of ingredients you can use in a recipe.

Range, on the other hand, is about the output values of a function. For a function, it's the set of all possible `y`-values you can get as a result. You can see it as the different dishes you can create with those ingredients.

So, in simple terms:

Domain: Inputs, or `x`-values

Range: Outputs, or `y`-values

Domain and Range of a Function

When we talk about a function `f`, where every element from a set `A` is linked to elements in another set `B`, `A` becomes the domain, and `B` is called the range. The function takes elements from the domain and transforms them into corresponding elements in the range.

For instance, if we have a function `f(x) = 2x`, the domain `D` could be all natural numbers, written as `{x: x ∈ N}`, and the range `R` would be all natural numbers too, but specifically the ones resulting from doubling each number in the domain.

When we represent a discrete function `f(x)` using a set of ordered pairs, the domain is the set of all `x`-coordinates. And range is the set of all `y`-coordinates.

For instance, if the function is `f(x) = {(1,10), (2, 20), (3, 30), (4, 40), (5, 50), (6, 30)}`, then:

• Domain `=` the set of all `x`-coordinates `= {1, 2, 3, 4, 5, 6}`
• Range `=` the set of all `y`-coordinates `= {10, 20, 30, 40, 50}`

`3` Ways to Write Domain and Range

We can define a set of values or data in many different forms. There are three main ways we can write the domain and range of a function:

• Inequality Notation: Inequality notation represents a set of values using open inequality symbols (`<` or `>`) or closed inequality symbols (`<=` or `>=`). Closed inequality symbols are used to include the boundary values whereas open inequality symbols are used to exclude boundary values. To represent `2` sets of data we use "or" between `2` inequalities. 
• Interval Notation: It is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. 
• Set Builder Notation: It is also called “Set Notation”. Set notation represents all values of `x` or `y` such that certain conditions are met. It takes the form `{x|` statement about `x}` which is read as, “the set of all `x` such that the statement about `x` is true.” The braces `{}` are read as “the set of,” and the vertical bar `|` is read as “such that”.

Let's illustrate these through some examples. 

Example `1`: The domain of a function is all `x`-values greater than or equal to `2`. Write the domain of the function in inequality notation, interval notation, and set builder notation.

Solution:

Inequality notation `D: x >=2`

Interval notation `D: [2,∞)`

Set Builder notation `D: {x| x>=2}`

Example `2`: The range of a function is all `y`-values less than `-3` or greater than or equal to `2`. Write the range of the function in inequality notation, interval notation, and set builder notation.

Solution:

Inequality notation `R: y<-3 or y >=2`

Interval notation `R: (-∞,-3) \cup [2,∞)`

Set Builder notation `R: {y| y<-3 or y >=2}`

Example `3`: The domain of a function is all `x`-values between `5` and `10`, including `5` and excluding `10`. Write the domain of the function in inequality notation, interval notation, and set builder notation.

Solution:

Inequality notation `D: 5<= x < 10`

Interval notation `D: [5,10)`

Set Builder notation `D: {x| 5<= x < 10}`

Example `4`: The range of a function is all real numbers. Write the domain of the function in inequality notation, interval notation, and set builder notation.

Solution:

Inequality notation `R:` \( \mathbb{R} \)

Interval notation `R: (-∞,∞)`

Set Builder notation `R: {x| x∈ℝ}`

Understanding the Domain of a Function

We know the domain of a function refers to all the possible input values that the function can accept. In simpler terms, it's the set of values that you can plug into the function. Imagine you have a function like \( f(x) = x^2 \), the domain would include all real numbers because you can plug in any real number for \( x \) and get a valid output. However, some functions have restrictions on what values can be plugged in, like square roots not accepting negative numbers. So, the domain helps us understand what inputs are allowed for a function.

Below are some commonly used functions. Let’s see what the domain of their parent functions looks like:

• Linear Function: The domain is all real numbers (\( \mathbb{R} \)).
• Quadratic Function: The domain is all real numbers (\( \mathbb{R} \)).
• Square Root Function: For a square root function \( \sqrt{x} \), the domain is \( x \geq 0 \).
• Exponential Function: The domain of an exponential function is all real numbers (\( \mathbb{R} \)).
• Logarithmic Function: The domain of a logarithmic function is \( x > 0 \).
• Rational Function: To find the domain of a rational function \( y = f(x) \), ensure that the denominator is not equal to zero.
• Polynomial Function: The domain of any polynomial function, whether linear, quadratic, cubic, etc., is all real numbers (\( \mathbb{R} \)).

How to Find the Domain of a Function?

Step `1`. Identify any restrictions imposed by the function's definition, such as square roots, fractions, or logarithmic functions.

Step `2`. Ensure that the input values satisfy these restrictions to avoid undefined results.

Example `1`: Find the domain of the function: \( f(x) = \sqrt{4x - 5} \)

Solution:

To find the domain:

Set the radicand (\( 2x - 3 \)) greater than or equal to zero to avoid taking the square root of a negative number.

\( 2x - 3 \geq 0 \)

Next, solve for \( x \).

\( 2x \geq 3 \)

\( x \geq \frac{3}{2} \)

So, the domain is \( x \geq \frac{3}{2} \).

Understanding the Range of a Function

The range of a function refers to all the possible output values that the function can produce. In simpler terms, it's the set of values that you can get out of the function after plugging in various inputs. 

For example, if you have a function \( f(x) = x^2 \), the range would be all non-negative real numbers because squaring any real number results in a non-negative number.

Below are some commonly used functions. Let’s see what the range of their parent functions look like:

• Linear Function: The range is all real numbers (\( \mathbb{R} \)).
• Quadratic Function: For \( y = a(x - h)^2 + k \), if \( a > 0 \), the range is \( y \geq k \); if \( a < 0 \), the range is \( y \leq k \).
• Square Root Functions: The range is \( y \geq 0 \).
• Exponential Functions: The range is \( y > 0 \).
• Logarithmic Functions: The range is \( \mathbb{R} \).
• Rational Functions: Solve for \( x \) and ensure the denominator is not equal to zero.

How to Find the Range of a Function?

Step `1`. Determine any limitations imposed by the function's structure, such as square roots, fractions, or trigonometric functions.

Step `2`. Ensure that the output values adhere to these constraints to prevent undefined outcomes.

Example: Find the range of the function: \( f(x) = x^2 + 3 \)

Solution:

Notice that \( x^2 \) always yields a non-negative result, so the lowest value for \( f(x) \) occurs when \( x = 0 \).

\( f(0) = 0^2 + 3 = 3 \)

There is no upper limit for \( x^2 \), so \( f(x) \) can increase indefinitely as \( x \) increases.

Therefore, the range of \( f(x) \) is \( f(x) \geq 3 \).

Thus, the range is \( y \geq 3 \).

Solved Examples

Example `1`. Find the domain and range of the function \( f(x) = \frac{1}{x} \).

Solution:

Domain:

The function has a restriction where the denominator cannot be zero.
\( x \neq 0 \)
So, the domain is all real numbers except \( x = 0 \). We can also write it as `{x| x∈ℝ, x ≠ 0}`.

Range:

As \( x \) approaches positive infinity, \( f(x) \) approaches zero, and as \( x \) approaches negative infinity, \( f(x) \) approaches zero. Therefore, the range is all real numbers except zero. We can also write it as `(-∞,0) \cup (0,∞)`.

Example `2`. Find the domain and range of the function \( g(x) = \sqrt{4 - x^2} \).

Solution:

Domain:

The radicand (\( 4 - x^2 \)) must be non-negative for real solutions.
\( 4 - x^2 \geq 0 \)
\( x^2 \leq 4 \)
\( -2 \leq x \leq 2 \)
So, the domain is `[-2, 2]`.

Range:

Since the square root function returns non-negative values, the range is all real numbers greater than or equal to zero. So, the range is \( x \geq 0 \). We can also write the range in interval notation as `[0,∞)`.

Example `3`. Find the domain and range of the function \( h(x) = \log(x + 3) \).

Solution:

Domain:

The argument of any logarithm must be greater than zero.
So, \( x + 3 > 0 \)
\( x > -3 \)
So, the domain is \( x > -3 \), which can be written in interval notation as `(-3,∞)`.

Range:

Since logarithm functions return all real numbers, the range is all real numbers. We can also write it as `x∈ℝ`.

Example `4`. Find the domain and range of the function \( f(x) = \frac{1}{x^2 - 4} \).

Solution:

Domain:

The function has a restriction where the denominator cannot be zero.
\( x^2 - 4 \neq 0 \)
\( x \neq -2, 2 \)
So, the domain is all real numbers except \( x = -2, 2 \). We can also write it in set builder notation as `{x: x∈ℝ, x ≠ -2, x ≠ 2}`

Range:

Since the function approaches zero as \( x \) approaches positive or negative infinity, the range is all real numbers except zero. We can also write it as `{x:x∈ℝ, x ≠ 0}`.

Example `5`. Find the domain of the function \( g(x) = \frac{\sqrt{x}}{x + 1} \)

Solution:

The radicand (\( x \)) must be non-negative.
\( x \geq 0 \)
Additionally, the denominator cannot be zero.
\( x + 1 \neq 0 \)
\( x \neq -1 \)
So, the domain is \( x \geq 0 \) and \( x \neq -1 \).

We can combine the two conditions to say that the domain is \( x \geq 0 \). We can also write it as `{x: x >=0}`.

Practice Problems

Q`1`. Which of the following represents the domain of the function \( f(x) = \frac{1}{x - 2} \)?

1. `{x| x ≠ 0}`
2. `{x| x ≠ 2}`
3. `{x| x >= 2}`
4. `{x| x < 2}`

Answer: b

Q`2`. What is the range of the function \( g(x) = x^2 + 3 \)?

1. \( y \geq 0 \)  
2. \( y > 0 \)  
3. \( y \leq 3 \)  
4. \( y \geq 3 \)

Answer: d

Q`3`. Which of the following represents the domain of the function \( h(x) = \sqrt{x - 4} \)?

1. `[4,∞)`
2. `(4,∞)`
3. `(-∞,4]`
4. `(-∞,4)`

Answer: a

Q`4`. What is the range of the function \( f(x) = \frac{1}{x^2 + 1} \)?

1. \( y > 0 \)  
2. \( y \geq 0 \)  
3. \( y \leq 1 \)  
4. \( y \geq 1 \)

Answer: c

Q`5`. Which of the following represents the domain of the function \( g(x) = \log(x^2 - 9) \)?

1. `{x|x > 3}`
2. `{x | x<=-3}`
3. `{x | x> -3}`  
4. `(-∞,-3) \cup (3,∞)`

Answer: d

Frequently Asked Questions

Q`1`. What is the domain of a function?

Answer: The domain of a function represents all the possible input values for that function. It's the set of values that you can plug into the function.

Q`2`. How do you find the domain of a function?

Answer: To find the domain of a function, you typically look for any restrictions or limitations on the input values so that the function is always defined. This may involve avoiding division by zero, ensuring that square roots operate on non-negative numbers, and considering any other restrictions imposed by the function's definition.

Q`3`. What is the range of a function?

Answer: The range of a function represents all the possible output values that the function can produce. It's the set of values that you can get out of the function.

Q`4`. How do you find the range of a function?

Answer: Finding the range of a function involves understanding how the function behaves as the input values vary. This may include identifying maximum and minimum values, considering asymptotic behavior, and accounting for any restrictions on the output values.

Q`5`. Why is it important to understand domain and range?

Answer: Understanding domain and range is crucial for various mathematical applications. It helps determine which inputs are valid for a given function, what outputs can be expected, and how the function behaves overall. This knowledge is essential for graphing functions, solving equations, and analyzing mathematical models in various fields.