# Exponential Function Reference

This is the general Exponential Function (see below for e^{x}):

f(x) = a^{x}

**a** is any value greater than 0

### Properties depend on value of "a"

- When
**a=1**, the graph is a horizontal line at**y=1** - Apart from that there are two cases to look at:

**a** between 0 and 1

Example: **f(x) = (0.5) ^{x}**

For **a** between 0 and 1

- As
**x**increases,**f(x)**heads to 0 - As
**x**decreases,**f(x)**heads to infinity - It is a Strictly Decreasing function (and so is "Injective")
- It has a Horizontal Asymptote along the x-axis (y=0).

**a** above 1

Example: **f(x) = (2) ^{x}**

For **a** above 1:

- As
**x**increases,**f(x)**heads to infinity - As
**x**decreases,**f(x)**heads to 0 - it is a Strictly Increasing function (and so is "Injective")
- It has a Horizontal Asymptote along the x-axis (y=0).

Plot the graph here (use the "a" slider)

### In General:

- It is
**always greater than 0**, and never crosses the x-axis - It always intersects the y-axis at y=1 ... in other words it passes through
**(0,1)** - At
**x=1**,**f(x)=a**... in other words it passes through**(1,a)** - It is an Injective (one-to-one) function

Its Domain is the Real Numbers:

Its Range is the Positive Real Numbers: (0, +∞)

## Inverse

a^{x} is the inverse function of log_{a}(x) (the Logarithmic Function)

So the Exponential Function can be "reversed" by the Logarithmic Function.

## The Natural Exponential Function

This is the **"Natural**" Exponential Function:

f(x) = e^{x}

Where **e** is "Eulers Number" = **2.718281828459... etc**

Graph of **f(x) = e ^{x}**

The value * e* is important because it creates these useful properties:

At any point the slope of **e**^{x} equals the value of **e**^{x} :

when x=0, the value of **e**^{x} = * 1*, and

**slope = 1**when x=1, the value of

**e**^{x}=

*, and*

**e**

**slope = e**etc...

The area **up to** any x-value is also equal to *e*^{x} :