Introduction to Exponential Functions

An <strong>exponential function</strong> has a variable in the exponent , this is what makes it fundamentally different from polynomial functions like $x^{2}$ or $x^{3}$ .

Exponential Function: A function of the form $f (x) = a^{x}$ , where the base $a$ is a positive constant with $a > 0$ and $a \neq = 1$ .

The base $a$ controls the function's behaviour:

If $a > 1$ , the function grows exponentially (e.g. $f (x) = 2^{x}$ )
If $0 < a < 1$ , the function decays exponentially (e.g. $g (x) = (0.5)^{x}$ )

Note that $a = 1$ is excluded because $1^{x} = 1$ for all $x$ , that's just a constant function, not interesting!

Analogy

Think of exponential growth like a rumour spreading at school. Each person tells two friends, who each tell two more , the number of people who know doubles every round. That's $2^{n}$ after $n$ rounds. No polynomial function grows this explosively.

Properties of the Parent Exponential Function

The parent exponential function $f (x) = a^{x}$ has several key properties you must know cold for the exam.

Property	Value
Domain	All real numbers, $x \in R$
Range	All positive reals, $y > 0$
$y$ -intercept	$(0, 1)$ , always, since $a^{0} = 1$
Horizontal asymptote	$y = 0$ (the $x$ -axis)
Type	One-to-one function

The function is always positive , no matter what you substitute for $x$ , $a^{x}$ can never be zero or negative.

Note

The natural exponential function $f (x) = e^{x}$ is a special and extremely important case, where the base is Euler's number $e \approx 2.71828$ . It appears throughout calculus, statistics, and real-world modelling. Its defining property is that its rate of change equals itself: $\frac{d}{d x} (e^{x}) = e^{x}$

Warning

A common error is confusing $f (x) = a^{x}$ (exponential) with $f (x) = x^{a}$ (power function). In the exponential, the variable is in the exponent. In a power function, the variable is the base. They behave very differently!

Properties of the Parent Exponential Function

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Property

Value

Domain

All real numbers,

x \in R

Range

All positive reals,

y > 0

y

-intercept

(0, 1)

, always, since

a^{0} = 1

Horizontal asymptote

y = 0

(the

x

-axis)

Type

One-to-one function

SL 2.9—Exponential and logarithmic functions

Introduction to Exponential Functions

Properties of the Parent Exponential Function

11 more sections in this topic

SL 2.9—Exponential and logarithmic functions

Introduction to Exponential Functions

Properties of the Parent Exponential Function

11 more sections in this topic