Exponential Functions: Definition, Formula and Examples

Exponential functions are mathematical functions in the form f(x) = a ⋅ b^x, where:

• a is a constant called the coefficient, which scales the function but does not change its exponential nature.
• b is the base of the exponential function, which must be a positive real number other than 1.
• x is the exponent, which is typically a variable.

Exponential functions are special types of mathematical functions that increase or decrease very quickly. The rate at which they grow or shrink depends on the value of the function itslef. This makes them perfect for describing things that grow or shrink rapidly, such as populations, money investments, or the way radioactive materials break down.

Key features of exponential functions include:

• If b > 1, the function exhibits exponential growth; it increases rapidly as x increases.
• If 0 < b < 1, the function shows exponential decay; it decreases rapidly as x increases.
• The graph of an exponential function never touches the x-axis but gets arbitrarily close to it as x approaches negative infinity.

Some examples of exponential functions are:

• f(x) = 2^x
• f(x) = 11^x
• f(x) = e^x/2
• f(x) = e^(-1/2x)

Exponential Function Formula

The formula of the exponential function is given as follows:

f(x) = a^x

where a > 0 and a ≠ 1 and x ∈ R

The most common exponential function is f(x) = e^x, where 'e' is "Euler's Number" and its value is "e = 2.718...."

The exponential curve depends on the exponential function and it depends on the value of the x. The exponential function is classified into two types based on the growth or decay of an exponential curve:

Exponential Growth

In Exponential Growth, a quantity increases slowly and then progresses rapidly. An exponentially growing function has an increasing graph. It can be used to illustrate economic growth, population expansion, compound interest, growth of bacteria in a culture, population increases, etc.

The formula for exponential growth is:

y = a(1 + r)^x

Where r is the Growth Percentage

Exponential Decay

In Exponential Decay, a quantity decreases very rapidly at first and then fades gradually. An exponentially decaying function has a decreasing graph. The concept of exponential decay can be applied to determine half-life, mean lifetime, population decay, radioactive decay, etc.

The formula for exponential decay is:

y = a(1 - r)^x

Where r is the Decay Percentage

Exponential Function Graph

The image given below represents the graphs of the exponential functions y = e^x and y = e^-x. From the graphs, we can understand that the e^x graph is increasing while the graph of e^-x is decreasing. The domain of both functions is the set of all real numbers, while the range is the set of all positive real numbers.

For an exponential function y = a^x (a > 1), the logarithm of y to base e is x = log_ay, which is the logarithmic function. Now, observe the graph of the natural logarithmic function y = log_ex. From the graph, we can notice that a logarithmic function is only defined for positive real values.

Exponential Series

Exponential function e^x can be expressed as an infinite series. This series is known as its Taylor series expansion, and it is derived using the function's derivatives evaluated at zero.

Exponential function e^x is defined as:

e^x = ∑ xⁿ/n!

Where n! (n factorial) is the product of all positive integers up to n.

Derivation of Exponential Series

Consider the exponential function f(x) = e^x

We know that, d/dx (e^x) = d²/dx²(e^x) = ... = dⁿ/dxⁿ(e^x) = e^x

Taylor series expansion of a function f(x) about x = 0 is given by:

f(x) = f(0) + f′(0).x/1! + f′′(0).x²/2! + f′′′(0).x³/3! + ⋯ ...(i)

f(x) = ∑ {f⁽ⁿ⁾(0)​/n!}xⁿ

f(x) = e^x, we have:
f(0) = e⁰ = 1
f′(0) = e⁰ = 1
f′′(0) = e⁰ = 1
f′′′(0) = e⁰ = 1
...

All derivatives of e^x evaluated at x = 0 are equal to 1.

Substituting these values into the Taylor series formula eq(i)

e^x = 1 + x/1!​ + x²/2!​ + x³/3! ​+ ⋯

Thus, the exponential function series is derived.

Exponential Function Properties

Below are some properties of the Exponential Function.

Domain and Range

• Domain: The domain of an exponential function is all real numbers, (−∞, ∞)
• Range: The range depends on the value of a. If a > 0, the range is (0, ∞). If a<0, the range is (−∞, 0).

Intercepts

• Y-intercept: When x = 0, f(0) = a⋅b⁰ = a. Thus, the y-intercept is at (0, a).
• X-intercept: For the standard exponential function a⋅b^x, there is no x-intercept because b^x ≠ 0 for any real x.

Asymptotes

• Exponential functions have a horizontal asymptote. For f(x) = a⋅b^x, if a > 0, the horizontal asymptote is y = 0. If a < 0, the horizontal asymptote is also y = 0, but the function approaches it from below.

Domain and Range of Exponential Functions

For a typical exponential function of the form, f(x) = ax, where a is a positive constant, the domain encompasses all real numbers. This means that you can input any real number x into the function.

On the other hand, the range of an exponential function is limited to positive real numbers. No matter what real number you choose for x, the output of f(x) will always be greater than zero. This is because any positive number raised to a power, whether that power is positive, negative, or zero, will result in a positive number.

Thus, the range of f(x) = a^x is (0, ∞), indicating that the function never touches or crosses the x-axis but grows indefinitely as x increases.

Exponential Graph of f(x) = 2^x

Let us consider an exponential function f(x) = 2^x.

From the graph of 2^x added above, we can observe that the graph of f(x) = 2^x is upward-sloping, increasing faster as the value of x increases. The graph formed is increasing and is also smooth and continuous. The graph lies above the X-axis and passes through (0, 1).

As x approaches negative infinity, the graph becomes arbitrarily close to the X-axis. The domain of an exponential function holds all real values, whereas its range contains all values greater than zero (y>0).

Exponential Function Rules

Following are some of the important formulas used for solving problems involving exponential functions:

Exponential Function Derivative

The derivative of the exponential function is added below.

For f(x) = e^x its derivative is,

d/dx (e^x) = e^x

For f(x) = a^x its derivative is,

d/dx (a^x) = a^x · ln a

Exponential Function Integration

The integration of the exponential function is added below.

For f(x) = e^x its integration is,

∫e^x dx = e^x + C

For f(x) = e^x its integration is,

∫a^x dx = a^x / (ln a) + C

Related Reads:

• Exponential Function Formula
• Derivative of Exponential Function
• Exponential Graph

Solved Examples of Exponential Functions

Example 1: Simplify the exponential function 5^x - 5^x+3.

Solution:

Given exponential function: 5^x - 5^x+3

From the properties of an exponential function, we have a^x × a^y = a^{(x + y)}

So, 5^x+3 = 5^x × 5³ = 125×5^x

Now, the given function can be written as

5^x - 5^x+3 = 5^x - 125 × 5^x
= 5^x(1 - 125)
=5^x(-124)
= -124(5^x)

Hence, the simplified form of the given exponential function is -124(5^x).

Example 2: Find the value of x in the given expression: 4³× (4)^x+5 = (4)^2x+12.

Solution:

Given, 4³× (4)^x+5 = (4)^2x+12

From the properties of an exponential function, we have a^x × a^y = a^{(x + y)}

⇒ (4)^3+x+5 = (4)^2x+12
⇒(4)^x+8 = (4)^2x+12

Now, as the bases are equal, equate the powers.

⇒ x + 8 = 2x + 12
⇒ x - 2x = 12 - 8
⇒ - x = 4
⇒ x = -4

Hence, the value of x is -4.

Example 3: Simplify: (3/4)^-6 × (3/4)⁸.

Solution:

Given: (3/4)^-6 × (3/4)⁸

From the properties of an exponential function, we have a^x × a^y = a^{(x + y)}

Thus, (3/4)^-6 × (3/4)⁸ = (3/4)^(-6+8)
= (3/4)²
= 3/4 × 3/4 = 9/16

Hence, (3/4)^-6 × (3/4)⁸ = 9/16.

Example 4: In the year 2009, the population of the town was 60,000. If the population is increasing every year by 7%, then what will be the population of the town after 5 years?

Solution:

Given data:

• Population of the town in 2009 (a) = 60,000
• Rate of increase (r) = 7%
• Time span (x) = 5 years

Now, by the formula for the exponential growth, we get,

y = a(1+ r)^x
= 60,000(1 + 0.07)⁵
= 60,000(1.07)⁵
= 84,153.1038 ≈ 84,153.

So, the population of the town after 5 years will be 84,153.

Exponential Functions Practice Questions

Question 1: Calculate the value of f(x) for f(x) = 3.2^x when x = 4.

Question 2: Given the exponential function g(x) = 5.(0.5)^x, sketch the graph of the function. Indicate the behavior of the function as x increases and as x decreases. Identify any asymptotes and intercepts.

Question 3: A population of bacteria doubles every hour. If the initial population is 200 bacteria, express the population P as an exponential function of time t in hours. Then, find the population after 6 hours.

Question 4: Solve for x in the exponential equation 10.3^x = 90.