June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function calculates an exponential decrease or increase in a certain base. For instance, let's say a country's population doubles every year. This population growth can be portrayed as an exponential function.

Exponential functions have numerous real-world use cases. Mathematically speaking, an exponential function is shown as f(x) = b^x.

Here we will learn the fundamentals of an exponential function in conjunction with relevant examples.

What is the equation for an Exponential Function?

The general formula for an exponential function is f(x) = b^x, where:

b is the base, and x is the exponent or power.
b is fixed, and x varies

As an illustration, if b = 2, we then get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x.

In cases where b is greater than 0 and does not equal 1, x will be a real number.

How do you chart Exponential Functions?

To chart an exponential function, we have to locate the spots where the function intersects the axes. This is referred to as the x and y-intercepts.

As the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.

To locate the y-coordinates, its essential to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

In following this approach, we achieve the domain and the range values for the function. Once we determine the worth, we need to plot them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share identical properties. When the base of an exponential function is more than 1, the graph will have the following qualities:

The line intersects the point (0,1)
The domain is all positive real numbers
The range is greater than 0
The graph is a curved line
The graph is rising
The graph is level and constant
As x approaches negative infinity, the graph is asymptomatic concerning the x-axis
As x advances toward positive infinity, the graph grows without bound.

In situations where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following qualities:

The graph crosses the point (0,1)
The range is greater than 0
The domain is all real numbers
The graph is decreasing
The graph is a curved line
As x advances toward positive infinity, the line in the graph is asymptotic to the x-axis.
As x gets closer to negative infinity, the line approaches without bound
The graph is smooth
The graph is unending

Rules

There are some basic rules to remember when engaging with exponential functions.

Rule 1: Multiply exponential functions with the same base, add the exponents.

For instance, if we need to multiply two exponential functions that posses a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with an equivalent base, subtract the exponents.

For instance, if we have to divide two exponential functions that posses a base of 3, we can write it as 3^x / 3^y = 3^(x-y).

Rule 3: To raise an exponential function to a power, multiply the exponents.

For example, if we have to grow an exponential function with a base of 4 to the third power, we are able to note it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is consistently equivalent to 1.

For example, 1^x = 1 regardless of what the rate of x is.

Rule 5: An exponential function with a base of 0 is always equivalent to 0.

For instance, 0^x = 0 no matter what the value of x is.

Examples

Exponential functions are usually used to signify exponential growth. As the variable grows, the value of the function grows quicker and quicker.

Example 1

Let’s observe the example of the growth of bacteria. Let’s say we have a group of bacteria that doubles hourly, then at the close of the first hour, we will have double as many bacteria.

At the end of hour two, we will have quadruple as many bacteria (2 x 2).

At the end of hour three, we will have 8x as many bacteria (2 x 2 x 2).

This rate of growth can be displayed utilizing an exponential function as follows:

f(t) = 2^t

where f(t) is the number of bacteria at time t and t is measured hourly.

Example 2

Also, exponential functions can represent exponential decay. Let’s say we had a dangerous substance that decomposes at a rate of half its quantity every hour, then at the end of one hour, we will have half as much substance.

At the end of hour two, we will have a quarter as much substance (1/2 x 1/2).

After hour three, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).

This can be shown using an exponential equation as follows:

f(t) = 1/2^t

where f(t) is the quantity of material at time t and t is assessed in hours.

As shown, both of these examples use a comparable pattern, which is the reason they can be shown using exponential functions.

As a matter of fact, any rate of change can be indicated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base stays constant. Therefore any exponential growth or decline where the base changes is not an exponential function.

For instance, in the matter of compound interest, the interest rate continues to be the same whereas the base varies in ordinary amounts of time.

Solution

An exponential function can be graphed using a table of values. To get the graph of an exponential function, we must enter different values for x and then measure the matching values for y.

Let's look at the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

To begin, let's make a table of values.

As you can see, the values of y rise very rapidly as x grows. Consider we were to draw this exponential function graph on a coordinate plane, it would look like the following:

As seen above, the graph is a curved line that rises from left to right ,getting steeper as it persists.

Example 2

Graph the following exponential function:

y = 1/2^x

First, let's make a table of values.

As you can see, the values of y decrease very quickly as x surges. The reason is because 1/2 is less than 1.

If we were to draw the x-values and y-values on a coordinate plane, it would look like what you see below:

The above is a decay function. As shown, the graph is a curved line that decreases from right to left and gets flatter as it goes.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be shown as f(ax)/dx = ax. All derivatives of exponential functions exhibit particular characteristics by which the derivative of the function is the function itself.

The above can be written as following: f'x = a^x = f(x).

Exponential Series

The exponential series is a power series whose terms are the powers of an independent variable number. The common form of an exponential series is:

Source

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