A mathematical function that contains exponents is called the **exponential function**. The term exponent is given to a value that denotes how many times a given number is to be multiplied by itself. It is also called the power or the degree of a number. For example, the exponent of x is 3 means the variable x is to be multiplied 3 times. So, an exponent indicates repeated multiplication of the same thing by itself. Here x is termed as the base and 3 is termed as the exponent.

The exponential function is an algebraic expression that involves variables as exponents. This means in an exponential function, one or more terms has a base that is constant but the exponent is a variable. Therefore, the exponential function of any variable is expressed by taking a constant as the base and the variable as the exponent.

The base of an exponential function that contains the variable as exponent should be positive and it should not be 1. For any exponential function having a base as 1, the value of the function becomes always 1 for any variable exponent. Thus the expression will become linear so it canâ€™t be called an exponential function.

An exponential function is used in many real-life situations and in many scientific and financial calculations to determine exponential growth or decay, return on investments, biological growth, etc. An exponential function refers to a situation where the quantity changes slowly at the beginning and then the rate of change increases and becomes faster over time. The exponential functions are helpful references for finding the growth or reduction in money, price, population, etc., that change exponentially over a while. Any quantity that grows or falls at the fixed rate at regular intervals can be expressed through an exponential function.

## Points of Exponential Functions

- If two exponential functions have the same base and their values are equal, then the exponents of both are equal to each other.
- An exponential function with a base of 1 becomes a linear function.

### Linear Function

A **linear function** is an algebraic expression that represents a straight line on the coordinate plane. It is a function in the form of an equation of a straight line that contains the variable with the coefficient which is called the value of the slope (the inclination of the line with the x-axis) plus the constant that indicates the y-intercept (where the line crosses the vertical y-axis). In other words, to represent a linear function, we have to know the value of the slope of the line and the intercept value where the line cuts the y-axis.

A linear function involving variable x can be expressed in the form of f(x) = y = mx + b

Here m denotes the slope of the line, b denotes the y-intercept of the line. Here x is called the independent variable and y is called the dependent variable as its value depends on the value of x.

For example, y = 2x + 9 is a linear function indicating a straight line with the slope 2 and y-intercept 9.

Again y = 5 is a linear function with slope zero and y-intercept 5.

Y =5x is another linear function that passes through the origin (0, 0)

Another form of a linear function that represents a straight line is by using the value of the slope, and the x-coordinate and y-coordinate of any point on that line. This can be written as follows:

y – y1 = m (x – x1)

Here, m is the slope of the line, x and y are variables that indicate coordinates of any point on the line, and x1 and y1 are the coordinates of a fixed point on the line.

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