When it comes to mathematics, there are many things that one can find challenging. The most basic and commonly used term is **equations**. We know what equations are from a basic level. However, one of the confusion that arises is how to know if an equation is a function. Most people usually use these terms interchangeably. We will revise both of them and then see how they differ, step by step.

First, using the terms equation and function interchangeably must be corrected, mainly when referring to the former. After all, whereas all parts are equations, not all equations are functions. Consequently, there is a chance of getting it wrong when referring to an equation as a function.

## 1. What Is a Function?

A mathematical function involves assigning an input value and getting a corresponding output value. Each input value should give a different output value to be considered a function. There are various functions in the real world that one can relate to.

An excellent example is the cost of fuel, which depends on the quantity one buys. Therefore, the prices will differ if two people buy the same quantity. On the other hand, two people who believe different quantities canโt be charged the same amount.

An ATM will also ensure that every user gets the amount requested. You can withdraw 3000 and expect an amount less or greater than that. It is an excellent example of a function that equates the value of x to that of y. If the input is 2000, the output will also be 2000.

If two vehicles travel the same distance, their time on the journey depends on their speed. If they travel at the same rate, the time spent will be the same and vice versa.

Lastly, taxi services that charge according to the distance or time also use a function to determine the respective charges for different passengers. The less the time or space, the less the costs, and vice versa. Only passengers who travel for the same time or distance will pay the same amount.

## 2. Various Types of Functions

In Mathematics, there are various types of functions. The most common types are as follows;

**Inverse function:**As the name suggests, this function can invert another one.**Polynomial function:**It is a function comprising polynomials.**Surjective function:**It is also referred to as the onto function. Such a function has two or more elements mapped from its domain to the range.**Injective function:**It is also referred to as a one-to-one function. It maps a particular range for every domain between a pair of sets.

A function is often written as an **algebraic equation**, and the expressions use corresponding operators, including exponentiation, multiplication, and addition.

Nevertheless, some functions are also expressed in other ways.

## 3. What Is an Equation?

An equation in mathematics is a statement you can use to solve a problem. It shows that two expressions are equal. In other words, whatever is on the right side of the equals sign should be equivalent to whatโs on its left side.

So, if you have an equation that says x + 5 = 10, it is a solution to finding the value of x. After all, adding 5 to get 10 is the value of x. Consequently, the sum of the left side is equivalent to the value of the elements on the right.

However, equations can be more complex, involving several variables. As long as two or more variables have different labels, they canโt have the same value.

## 4. How Do You Determine if an Equation Is a Function?

If you want to determine whether an equation is a function, follow this process.

### 4.1 Identifying Its Input Values

A function has an input value. Any input value of a process is a variable, but you are the one who assigns it its value. Thatโs why the input values are usually called independent valuables.

### 4.2 Identifying Its Output Values

A function also has an output value, which is usually one. It is also a variable but termed dependent since it depends on the input values. In other words, the value of your input values will determine the value of the output.

### 4.3 Establishing the Relationship between the Input and Output Values of the Equation

For a part, you can only have a single output for every value you assign the input. Therefore, if you have an equation with two results for a particular input value, thatโs not a function.

The bottom line is that an equation must satisfy these three properties to be considered a function;

- Each variable should appear a minimum of one time
- There should be a dependent variable depending on the independent one
- Lastly, the equation can only have one output value or solution

## 5. Examples of Equations Which Are Functions

If you have an equation that says y = 2x + 5, it is easy to determine whether it is a function. At a glance, y is the output value or the dependent value. On the other hand, x is its input value or independent value.

So, let us assign 3 to the x variable. Under such circumstances, y will equal 2 x 3 + 5 = 11. If we give the value of x to 2, the value of y changes to 2 x 2 + 5 = 9.

Every value of x gives a different value of y. Therefore, y = this equation is a function.

## 6. Examples of Equations Which Arenโt Functions

For example, if y is the square root of x, that canโt be a function. After all, any numberโs square root can be its positive and negative integer. The square root of 4 can be -2 or 2. Since these are two output values for the same input value, it isnโt a function despite being an equation.

If an equation forms a vertical line, it canโt be a function. After all, that would mean that a value of its input value would have more than one possible output value. Such a relationship isnโt a characteristic of an equation thatโs a function.

## Conclusion

A function is an equation, but its characteristics distinguish it from other equations.

A function has an independent value and a dependent one. Thatโs usually an input value and an output value. The value you assign the input will affect the output. Consequently, as long as the inputs are different, the results canโt be the same.

Last Updated on by Sathi

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