Exponential Function

Maths: Edexcel A Level Pure Maths, topic 6.1.1

Exponential Functions

Functions that are linear show growth or decay that occurs at a steady rate. The exponential function models growth or decay with an increasing rate.

Definition

Definition

For any real number $x$ , an exponential function is a function with the form"
$f(x) = ab^x$
Where $a$ is a non-zero real number called the initial value and $b$ is any positive real number such that $b\ne1$ .
We call $b$ the base of the exponential function.
$f(x) = 3^x$ has base 3.

Exponential growth

Exponential growth

Exponential growth means that the original value increases by the same percentage over equal increments found in the domain.
As the value grows, so does the percentage of the value.
Linear growth means the original value increases by the same amount over equal increments found in the domain.

Exponential growth

Exponential growth

The table shows the values for which the exponential function of base 2 is larger than linear and quadratic functions.
In general, the exponential function will always eventually outgrow any polynomial.

Graph

Graph

The $y$ -intercept of an exponential function is $(0,a)$ .
For a function with no initial value such as $f(x)=2^x$ , this intercept is always equal to 1.
The graph does not cross the $x$ -axis, instead getting closer toward it for very large and negative values of $x$ .

Properties

Properties

The domain of $f(x) = ab^x$ is all of the real numbers.
The range is all positive real numbers if $a>0$ .
The range is all negative real numbers if $a<0$ .

Test yourself on Exponential Function

What is the domain of the function $s(n) = 6^n$ ?
What is the range of the function $b(x) = -10^x$ ?

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