An exponential function is a mathematical function used in many real-world situations. It is mainly used to find exponential decay or exponential growth, compute investments, model populations, and more. In this article, you will learn about exponential function formulas, rules, properties, graphs, derivatives, exponential series, and examples.

An exponential function, as its name suggests, involves exponents. It has a constant as its base and a variable as its exponent. An exponential function can be in one of the following forms.

In mathematics, an exponential function is a function of the form f(x) = a^x, where "x" is a variable and "a" is a constant called the base of the function. The base "a" must be greater than 0.

Here are some examples of exponential functions:

• f(x) = 2^x
• f(x) = (1/2)^x
• f(x) = 3e^2x
• f(x) = 4 (3)^-0.5x

An exponential function is defined by the formula f(x) = a^x, where the input variable x occurs as an exponent. The exponential curve depends on the exponential function and the value of x.

The exponential function is an important mathematical function of the form:

f(x) = a^x

Where:
-a > 0 and a ≠ 1
- x is any real number
- If the variable is negative, the function is undefined for -1 < x < 1

An exponential curve grows or decays depending on the exponential function. Any quantity that grows or decays by a fixed percentage at regular intervals exhibits exponential growth or decay.

Exponential Growth:
In exponential growth, the quantity increases slowly at first and then rapidly. The rate of change increases over time. The formula for exponential growth is:

y = a (1 + r)^x

Where r is the growth percentage.

Exponential Decay:
In exponential decay, the quantity decreases rapidly at first and then slowly. The rate of change decreases over time. The formula for exponential decay is:

y = a (1 - r)^x

Where r is the decay percentage.

The following figure represents the graph of exponents of x. As the exponent increases, the curves get steeper, and the rate of growth increases. Thus, for x > 1, the value of y = f_n(x) increases for increasing values of n.

The exponential function with a base greater than 1, i.e., a > 1, is defined as y = f(x) = a^x. The domain of the exponential function is the set of all real numbers R, and the range is the set of all positive real numbers.

The graph of an exponential function passes through the point (0, 1) and is increasing. It is very close to zero if the value of x is mostly negative.

The natural exponential function is y = e^x, where e is the base of the natural logarithm.

The properties of the exponential function graph when the base is greater than 1 are:

• The graph passes through the point (0, 1).
• The domain is all real numbers.
• The range is y > 0.
• The graph is increasing.
• The graph is asymptotic to the x-axis as x approaches negative infinity.
• The graph increases without bound as x approaches positive infinity.
• The graph is continuous.
• The graph is smooth.

For the exponential function y = 2^-x, the properties are:

• The graph passes through the point (0, 1).
• The domain is all real numbers.
• The range is y > 0.
• The graph is decreasing.
• The graph is asymptotic to the x-axis as x approaches positive infinity.
• The graph increases without bound as x approaches negative infinity.
• The graph is continuous.
• The graph is smooth.

Some important exponential rules are:

• a^x a^y = a^{x + y}
• a^x / a^y = a^{x - y}
• (a^x)^y = a^xy
• a^x b^x = (ab)^x
• (a / b)^x = a^x / b^x
• a⁰ = 1
• a^-x = 1 / a^x

Examples of exponential functions include:

• f(x) = 2^x
• f(x) = (1/2)^x
• f(x) = 2^{x + 3}
• f(x) = 0.5^x