Contents

**Image|Definition & Meaning**

**Definition**

In mathematics, the **image** is an **equivalent** term for the **range** of a **function**. Therefore, the image of a function is the **set** of **all** the **possible** **output** **values** of a **function**. Simply put, the domain of a function “maps” to the image (or range) of a function.

Figure 1 – Illustration of the image of a function

The **range** of a **function**, also referred to as the **image** of the **function**, is the collection of all **possible outputs.** A function’s image is significant because it provides a **visual** **representation** of the **function’s** **output** values, enabling us to **spot** **patterns** and **make** **predictions**. The idea of the image of a function and its uses in mathematics and practical issues will be discussed in this article.

**Demonstrating the Concept of Image**

In mathematics, we designate the **image** of a **function** as **f(X)**, where **X** is the **set** of all **potential** **inputs**, and the **function** f is one that **maps** **inputs** x to **outputs** y. The **image** is a **subset** of the **codomain**, which is the **set** of all **possible** **outputs** and is **referred** to as the **codomain** of the function.

**Consider** the function** f(x) = x ^{2}** as an illustration. Any

**real**

**number**may be used as an

**input**

**value**(x), and the

**output value**(y)

**will**be the

**square**of the

**input**

**value**. The set of all feasible real number squares, also referred to as the set of

**non**–

**negative**

**real**

**numbers**, is the

**function**f’s

**image**.

**Visualizing Image**

Mathematicians often find it helpful to **visualize** the **image** of a function because it **clarifies** how the **function** **behaves** and reveals patterns in its results. The **graphical** **representation** of a **function** in mathematics frequently **places** the **inputs** on the **x-axis** and the **outputs** on the **y-axis**.

We **can** **see** how the **output** values **change** as the **input** values **change** by **plotting** the **image** of a function, and we can **spot** any **patterns** or **trends** in the **data**.

Consider the function **f(x) = x ^{2}** as an illustration.

**When**we

**plot**the function’s image, we

**obtain**a

**parabolic**

**shape**, as shown in figure below.

The **image** of the function, as shown by the points on the graph, **is** a **collection** of **non-negative** **real** **numbers**. Additionally, we can see that the **function** is **increasing**, which means that as the input values rise, so do the output values.

**Properties of Image**

Some properties of a function’s image are detailed below:

- A function’s
**codomain**is**subset**by its**image**. **Depending**on how many**outputs**a function generates, its**image**can either be**finite**or**infinite**.- Depending on how the outputs are distributed, the
**image**of a function can either be**discrete**or**continuous**. - A function’s
**image**is always**closed**, which means it**includes**its**boundary****points**. - Depending on the kind of inputs and outputs, the image of a function can be a
**collection**of**real****numbers**, complex numbers, or any other kind of mathematical object. - No matter what the inputs are, a
**function’s**image is always**determined**by its**outputs**. **Finding**the**highest**and**lowest****outputs**of a function can**yield**its**image**, as can employing algebraic strategies like locating the function’s**domain**and**codomain**.- If the
**inputs**are**limited**to a smaller**set**or the**function**is**altered**in some way, the**image**of the function may**change**. - A
**graph**or**table**of values can be**used**to**depict**the**image**of a function.

These properties are extremely helpful when we get into more complex applications of mathematics.

**Finding an Image of a Function**

**Depending** on the **kind** of **function** and the kind of output, there are various ways to determine a function’s range(image). The most typical approach is to use the **function’s** **graph** to determine the range.

**Method 1**

Figure 2 – Finding an image of a parabola using geometry

We must ascertain the **highest** and **lowest** output values that the function generates in order to **determine** the **range**(image) of a function using its graph. Consider the **function** **f(x) = x ^{2}** as an illustration.

**Finding**the

**vertex**of the

**parabola**will

**reveal**the output values that are

**highest**and

**lowest**for this function, which has a parabolic graph.

The **vertex** is the **point** on the **parabola’s** **graph** where it reaches its highest or lowest point. It is also the **location** where the function’s **x-coordinate** **equals** the **average** of the **x-coordinates** of the two graph points with the **same y-coordinate**.

**Method 2**

Figure 3 – Finding the image of a function using the algebraic method

Using **algebraic** **methods**, such as **identifying** the **function’s** **domain** and **codomain** before identifying the **subset** of the **codomain** that **corresponds** to the **range**(image), is another way to determine a function’s range (image).

**Consider** the function **f(x) = x ^{2} + 4** as an illustration. We must first

**identify**the

**codomain**, which is the set of all real numbers, in order to determine the function’s range(image). The

**subset**of the

**codomain**that corresponds to the

**range**(image), or the

**set**of

**all**

**real**

**numbers**

**greater**than or

**equal**to

**4**, must then be located.

**Application of Image of Function**

In mathematics and numerous fields, the image (or range) of a function has several crucial applications, such as.

**Modeling real-world phenomena:**Real-world phenomena like population growth, supply and demand, or temperature change can be modeled using an image of a function.**Engineering and physics:**Relationships between variables, such as force, velocity, and acceleration, are described using the image of a function.**Economics:**To model market trends and forecast consumer behavior, economists use images of functions.**Control systems:**To design systems that can automatically modify their behavior based on inputs, control systems use an image of a function.**Data visualization:**Images of functions can be represented graphically as graphs, which offer a clear and concise illustration of the data and the connections between it.**Image and signal processing:**To analyze and manipulate digital signals, such as audio and video signals, the image of a function is used.

There are numerous other cases where knowing the image of a function allows us to make convenient assumptions and use them to model various problems.

**Example**

Consider the function** f(x) = x ^{2}**.

**Find**an

**image**of

**value**

**4**by this function.

**Solution**

Figure 4 – Example of an image of a function

f(x) = x^{2}

f(x) = (4)^{2}

f(x) = 16

So the **image** of 4 by the function f is **f(4) = 16**

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