Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function calculates an exponential decrease or increase in a particular base. Take this, for example, let us assume a country's population doubles yearly. This population growth can be depicted as an exponential function.

Exponential functions have numerous real-life applications. Expressed mathematically, an exponential function is written as f(x) = b^x.

In this piece, we discuss the fundamentals of an exponential function along with appropriate examples.

What’s the formula for an Exponential Function?

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

1. b is the base, and x is the exponent or power.

2. b is a constant, and x is a variable

For example, if b = 2, then we 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 a situation where b is larger than 0 and does not equal 1, x will be a real number.

How do you plot Exponential Functions?

To chart an exponential function, we have to find the dots where the function crosses the axes. These are known as the x and y-intercepts.

Considering the fact that the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.

To find the y-coordinates, its essential to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2

In following this method, 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 comparable qualities. When the base of an exponential function is more than 1, the graph will have the following properties:

• The line crosses the point (0,1)

• The domain is all positive real numbers

• The range is larger than 0

• The graph is a curved line

• The graph is increasing

• The graph is smooth and constant

• As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis

• As x nears positive infinity, the graph increases without bound.

In events where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following properties:

• The graph passes the point (0,1)

• The range is more 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 within graph is asymptotic to the x-axis.

• As x advances toward negative infinity, the line approaches without bound

• The graph is level

• The graph is constant

Rules

There are several vital rules to bear in mind when dealing with exponential functions.

Rule 1: Multiply exponential functions with an identical 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 identical base, subtract the exponents.

For example, if we have to divide two exponential functions that have 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 raise an exponential function with a base of 4 to the third power, we are able to write it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is forever equal to 1.

For instance, 1^x = 1 regardless of what the value of x is.

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

For example, 0^x = 0 regardless of what the value of x is.

Examples

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

Example 1

Let’s observe the example of the growth of bacteria. Let’s say we have a culture of bacteria that multiples by two hourly, then at the end of the first hour, we will have twice as many bacteria.

At the end of the second hour, we will have 4x as many bacteria (2 x 2).

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

This rate of growth can be represented using an exponential function as follows:

f(t) = 2^t

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

Example 2

Moreover, exponential functions can portray exponential decay. If we have a dangerous substance that decomposes at a rate of half its amount every hour, then at the end of one hour, we will have half as much substance.

After the second hour, we will have one-fourth as much material (1/2 x 1/2).

At the end of three hours, we will have an eighth as much material (1/2 x 1/2 x 1/2).

This can be displayed using an exponential equation as follows:

f(t) = 1/2^t

where f(t) is the volume of substance at time t and t is assessed in hours.

As you can see, both of these samples pursue a similar pattern, which is the reason they are able to be represented using exponential functions.

As a matter of fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable while the base continues to be the same. This means that any exponential growth or decline where the base varies is not an exponential function.

For instance, in the scenario of compound interest, the interest rate stays the same while the base varies in regular time periods.

Solution

An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we have to plug in different values for x and then measure the corresponding values for y.

Let's check out the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As shown, the worth of y grow very rapidly as x increases. 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 goes up from left to right and gets steeper as it continues.

Example 2

Draw the following exponential function:

y = 1/2^x

To start, let's create a table of values.

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

If we were to graph the x-values and y-values on a coordinate plane, it would look like the following:

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 proceeds.

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 special features whereby 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 terminology are the powers of an independent variable digit. The general form of an exponential series is:

Source

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