Exponential Functions| Definition, Formula, Examples

Exponential Function is said to be a function that involves exponents. An exponential function is classified into two types, i.e., exponential growth and exponential decay.

Let’s discuss exponential functions in detail, including its graph, types, formulas and solved examples.

Exponential Function Definition

A mathematical function in the form of f(x) = a^x, where “a” is called the base of the function, which is a constant greater than 0, and “x” is the exponent of the function, which is a variable. When x > 1, the function f(x) increases with increasing x values. The value of “e” is approximately equal to 2.71828.

Exponential Function Formula

The formula for an exponential function is given as follows:

f(x) = a^x, where a>0 and a ≠ 1 and x ∈ R

Exponential function is classified into two types based on the growth or decay of an exponential curve. They are :

1. Exponential Growth
2. Exponential Decay

Exponential Growth

In exponential growth, a quantity increases very slowly at first 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.

Read More :

• Exponential Growth Formula
• Exponential Decay Formula

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.

Exponential Curve

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.

As the logarithmic function is not defined for negative values, its domain is the set of all positive real numbers.

Curve comparison of log x and ex

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)=ax is (0,∞)(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.

x	-3	-2	-1	0	1	2	3
f(x)= 2x	f(-3) = 2-3 = 1/8 = 0.125	f(-2) = 2-2 = 1/4 = 0.25	f(-1) = 2-1 = 1/2 = 0.5	f(0) = 20 = 1	f(1) = 21 = 2	f(2) = 22 = 4	f(3)= 23 = 8

Graph of 2x

From the graph, 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 graph of f(x) = 2^-x

Let us consider an exponential function f(x) = 2^-x = (1/2)^x.

x	-3	-2	-1	0	1	2	3
f(x) = (1/2)x	f(-3) = (1/2)-3=8	f(-2)=(1/2)-2=4	f(-1)=(1/2)-1=2	f(0)=(1/2)0=1	f(1)=(1/2)1= 0.5	f(2)=(1/2)2= 1/4=0.25	f(3)=(1/2)3=1/8= 0.125

Graph of 2-x

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

Graph of 10x

The image given above represents the graph of exponents of x. One can observe that as the exponent of the function increases, the curve is getting steeper. So, we can conclude that the nature of a polynomial function depends on its degree. Therefore, for all x > 1, a function y = f_n(x) increases as the value of “n” increases.

Thus, any polynomial function with a higher degree has a higher growth. However, a function f(x) = a^x (where a > 1) grows faster than a polynomial function. Hence, for all positive integers n, the function f (x) grows faster than the function f_n (x).

Exponential Function Rules

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

Rules for Exponential Functions
Power of zero rule	a0 =1
Negative power rule	a-x = 1/ax
Product Rule	ax × ay = a(x + y)
Quotient Rule	ax/ay = a(x – y)
Power of power rule	(ax)y = axy
Power of a product power rule	ax × bx=(ab)x
Power of a fraction rule	(a/b)x= ax/bx
Fractional exponent rule	(a)1/y = y√a (a)x/y = y√(ax)

Related :

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

Exponential Function Examples with Answers

Let’s solve some questionas on the 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.

FAQs on Exponential Function

What is Exponential Function?

An exponential function is defined as a mathematical function with the formula f(x) = ax, where “x” is a variable and is known as the exponent of the function, and “a” is a constant greater than zero and is known as the base of the function.

What are properties of Exponential Function?

The domain of an exponential function holds all real values, whereas its range contains all values greater than zero (y>0). Its graph is upward-sloping, increasing faster as the value of x increases. 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.

Mention some Exponential Formulas.

The following are some exponential formulas for exponential functions. They hold true if a > 0 and for all real values of m and n.

• a^x × a^y = a^{(x + y)}
• a^x/ay = a^{(x – y)}
• a⁰ = 1
• a^-x = 1/a^x
• (a^m)ⁿ = a^mn

What are the types of Exponential Functions?

The exponential function is classified into two types based on the growth or decay of an exponential curve, i.e., exponential growth and exponential decay. As the name suggests, in exponential growth, a quantity increases very slowly at first and then progresses rapidly, while in exponential decay, a quantity decreases very rapidly at first and then fades gradually.

What are the formula of exponential growth and exponential decay?

The formula for exponential growth is:

y = a(1+ r)^x where r is the growth percentage.

The formula for exponential decay is:

y = a(1 – r)^x, where r is the decay percentage.