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# Range of a Function|Definition & Meaning

## Definition

The range of a function is the set of the output values a function actually produces for a given set of inputs (its domain). Thus, for a function f(x) = 2x + 1, if the domain is the set of all natural numbers (i.e., x $\in$ {1, 2, 3, …}), then the range is the set of all odd natural numbers except one since f(x={1, 2, 3, …}) = y = {3, 5, 7, …}.

If a person is interested in pursuing a career in **mathematics** or if one requires the methods to solve everyday problems in business, it becomes fairly important to understand and apply different **formulas** and **solutions** effectively.

If you are curious about finding the **range** of a particular **function**, there are numerous ways to carry out this operation, but it is more important that you must know about the basics of a **function** and its **domain** which results in the **range** of a **function**.

## What Is a Function?

Any sentence or a group of letters and numbers that you see having a relational sign in between is known as a **function**. The relational sign can be an equal to, less than, or greater than, and so forth. It basically tells you the exact **relationship** between two sets of identical or distinct variables.

The mathematical expression of a **function** looks something like a formula:

y = f(x)

In the above **expression**, the left-hand side represents the dependent variable, which is dependent on the **variability** of the expression on the right-hand side. Thus y can be described as a **function** of x, which means that whenever there is a slight change in the **value** of x, the **value** of y will correspondingly change depending on the structure of the **function**.

Here y is also known as the **range** of the **function**, allowing us to determine the extent of a **function**, whereas the **value** x represents the **domain**, which can be any arbitrary **value**.

For instance, the simplest **function** can be written as:

y = x – 1

If we take x = 2 and put it in the above equation, we get:

y = 2 – 1 = 1

Similarly, changing the **value** of x to 10 will result in y = 10 – 1 = 9.

## What Is Range?

As discussed above, the **range** of a **function** is the total extent to which the **function** can stand out. In simple words, a **function** requires a set of **domain** **values**, to predict the overall **range** of the **function**. We can define **domain** and **range** as,

### Domain

It is the set of **values** that are injected into a **function**, as an input. They represent the **values** of x in most cases.

### Range

It represents the outcome of a **function**, for every **value** of the input. In our case, y represents the **range** of the **function** based on every **value** of x.

In the above figure, the **function** is y = f(x) = x^{2}, which means that for every **value** of x, the **value** of y will double, thus if a set of numbers is provided to the **function**, let’s say {1,2,3,…}, it will give the **range** as the output, that is {1,4,9,…}.

## How To Find the Range of a Function?

If we are to work with an ordered pair of (x,y), the **value** of x will only correspond to one single **value** of y. But for y, there can be a number of possibilities. This means that we have to find the **values** of y based on the given set of **values** of x. We will be discussing three ways to find the **range**, by using a **formula**, a **graph**, and by using a **relationship**.

### By Using a Formula

The **relationship** between the variables x and y can be represented mathematically. Relying on the nature of the interactions between the **values**, these formulas can have various appearances. The procedures for finding a mathematical **function**‘s **range** are as follows,

### Write the Formula

The **formula** can give many aspects which help in determining the **relationship** between different variables. Such a formula can be y = f(x). Let’s say you sell tomatoes for 1$ each, so your total **sales** **depend** on the number of tomatoes sold multiplied by the cost of each tomato, making a formula f(x) = 1(x). If you sell a total of 10 tomatoes, our sales will be \$10, but if you sell only 1 tomato, your sale will be \$1.

### View More Coordinate Pairs

Since the sale can only be a positive **function**, you can go for more information by drawing **ordered** **pairs** on a graph. This will help you understand the trend, whether it is linear or upwards. This also helps to find the **relationship** between x and y.

### Write Down the Range

Since you have already figured out that your sales cannot go **negative**, the **range** of your sales will never be lower than **zero**. The reason is that your sale will always tend to increase instead of go down. As you know that the sales will increase by a factor of 1, so the **range** will be:

f(x) = for all multiples of 1 $ge$ 0

### By Using a Graph

A visual representation of a **function** can significantly help in determining the **relationship** of x and y. The procedure to determine the **range** using a graph is as follows,

### Draw the Graph of the Function

Draw the **function** on graph paper by marking the x and y **values** using small dots. This will help in visualizing the shape of the **function**, whether it’s a ‘u’ or ‘n’ or any arbitrary shape.

The next step is to find the **minimum**, which can be located at the lowest point on the graph.

Similarly, the maximum of a **function** can be located at the highest point on the graph.

### Figure Out the Range

The **range** can be always equal in relation to the **domain**, it may be **greater** than or **less** than a certain **value**. For instance, the **range** {-1,1,2,3}, can be stated as -1 $le$ f(x) $ge$ 3.

## Solved Example Using Range of a Function

For the **function** given below, determine the **domain** and **range:**

f(x) = 3x^{2} – 5

### Solution

We are given a **function** f(x) = 3x^{2} – 5

The **domain** of this **function** will be the set of **values** we provide as an input, for which we get the output as real and defined **values**. Since the **function** has no indefinite x **values**, the **domain** of the **function** is going to always be real and well-defined. Thus:

Domain = D = [-$\infty,\infty $]

Now for determining the **range** of the **function**, we have to find the **values** of y, which are dependent on the **values** of x given in the **function**. So:

y = 3x^{2} – 5

3x^{2} = y + 5

x^{2} = (y+5) / 3

x = $\mathsf{\sqrt{\dfrac{(y+5)}{3}}}$

Figure 3 – Graph of example problem

For this square root to be a positive real number, y must be greater than or equal to -5.

Thus, the **range** of this **function** is [-5, $\infty$)

*All images/mathematical drawings were created with GeoGebra.*