AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsSequences & Exponential FunctionsGeneral Exponential Functions

General Exponential Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

General exponential functions

What is the general form of an exponential function?

The general form of an exponential function is $f (x) = a b^{x}$
- $a$ is the initial value
  - i.e. the output when $x = 0$
    - since $f (0) = a b^{0} = a$
  - $a \neq 0$ (otherwise the function is just the constant zero)
- $b$ is the base
  - The base must satisfy $b > 0$ and $b \neq 1$

When $a > 0$ and $b > 1$ , the function demonstrates exponential growth
- the output values increase as $x$ increases
When $a > 0$ and $0 < b < 1$ , the function demonstrates exponential decay
- the output values decrease as $x$ increases
When $a < 0$ , the graph is reflected across the $x$ -axis compared to the corresponding positive $a$ case

Four graphs of y = ab^x showing different curves based on positive and negative values of a, and for both b>1 and 0<b<1. — **Graphs of y=ab^x for different values of a and b**

What is the domain of an exponential function?

The domain of an exponential function is all real numbers
When the inputs are natural numbers ( $1, 2, 3, \dots$ ), the input value specifies
- the number of factors of the base $b$ to apply to the initial value $a$
- E.g. for $f (x) = 5 \cdot 3^{x}$
  - $f (1) = 5 \cdot 3$
  - $f (2) = 5 \cdot 3 \cdot 3$
  - $f (3) = 5 \cdot 3 \cdot 3 \cdot 3$
  - etc.
- This connects directly to the idea of geometric sequences as repeated multiplication
However an exponential function is also defined for non-integer and negative inputs
- E.g. for $f (x) = 5 \cdot 3^{x}$
  - $f (0.5) = 5 \cdot 3^{0.5} = 5 \sqrt{3}$
  - $f (- 1) = 5 \cdot 3^{- 1} = \frac{5}{3}$

What key characteristics does an exponential function have?

The output values of an exponential function in general form are proportional over equal-length input-value intervals; therefore
- Exponential functions are always increasing or always decreasing
  - They do not change direction
- Their graphs are always concave up or always concave down
  - The concavity does not change
As a consequence:
- Exponential functions do not have extrema (i.e. maximum or minimum values)
  - except on a closed interval
- Their graphs do not have points of inflection

How can you identify a 'hidden' exponential function?

Sometimes data does not immediately appear to be exponential, because a constant has been added to the output values
If the output values of a function $f$ are not proportional over equal-length input-value intervals
- but the output values of an additive transformation $g (x) = f (x) + k$ are proportional over equal-length input-value intervals
- then $f$ can be modeled by an additive transformation of an exponential function
E.g. consider the data points $(1, 7)$ , $(2, 9)$ , $(3, 13)$ , $(4, 21)$
- The ratios of successive outputs are $\frac{9}{7}$ , $\frac{13}{9}$ , $\frac{21}{13}$
  - These are not constant, so the data does not look exponential at first
- But if you subtract 5 from each output value, you get $(1, 2)$ , $(2, 4)$ , $(3, 8)$ , $(4, 16)$
- Now the ratios are $\frac{4}{2} = 2$ , $\frac{8}{4} = 2$ , $\frac{16}{8} = 2$
  - I.e. they are constant
- So the original data can be modeled by $f (x) = 2^{x} + 5$
  - an exponential function with a vertical shift

What are the end behaviors of an exponential function?

The end behavior of an exponential function $a b^{x}$ depends on the values of $a$ and $b$
- This is summarised in the table below
- Note that changing $a > 0$ to $a < 0$ 'flips' the values of the unbounded limits

Values of $a$ and $b$	$\lim_{x \to - \infty} a b^{x}$	$\lim_{x \to \infty} a b^{x}$
$a > 0, b > 1$	0	$\infty$
$a > 0, 0 < b < 1$	$\infty$	0
$a < 0, b > 1$	0	$- \infty$
$a < 0, 0 < b < 1$	$- \infty$	0

Examiner Tips and Tricks

On the calculator part of an exam, you can use your graphing calculator to help you spot the behavior of a function as $x$ increases or decreases without bound.

But be sure you can analyze these behaviors without your calculator as well!

The natural base e

What is the natural base e?

The number $e$ is a special mathematical constant, approximately equal to 2.718
- Like $π$ , $e$ is an irrational number
  - Its decimal expansion goes on forever without repeating
  - $e = 2.7182818284590452353602874713527 . . .$
$e$ is often used as the base in exponential functions that model real-world phenomena
- For example, continuous growth and decay processes in science, finance, and other fields are often expressed using base $e$
An exponential function with base $e$ is written as $f (x) = a e^{x}$
- This follows the same general form $f (x) = a b^{x}$
  - with $b = e$
Since $e \approx 2.718 > 1$ , the function $f (x) = e^{x}$ (with $a = 1 > 0$ ) demonstrates exponential growth
You will encounter the natural base $e$ frequently in modeling contexts

Worked Example

$x$	$0$	$1$	$2$	$3$	$4$
$f (x)$	$36$	$18$	$9$	$\frac{9}{2}$	$\frac{9}{4}$

The exponential function $f$ is defined by $f (x) = a b^{x}$ , where $a$ and $b$ are positive constants. The table gives values of $f (x)$ at selected values of $x$ . Which of the following statements is true?

(A) $f$ demonstrates exponential decay because $a > 0$ and $0 < b < 1$ .

(B) $f$ demonstrates exponential decay because $a > 0$ and $b > 1$ .

(D) $f$ demonstrates exponential growth because $a > 0$ and $b > 1$ .

Answer:

The function values are decreasing as $x$ increases, which means that this is an example of exponential decay

That rules out options C and D

You should know that exponential decay corresponds to $0 < b < 1$ in the exponential formula given

However if you forgot that you could use the figures in the table to find the value of $b$
When $x = 0$ :

$a b^{0} = 36 \Rightarrow a \cdot 1 = 36 \Rightarrow a = 36$

Using that value of $a$ , then when $x = 1$ :

$36 \cdot b^{1} = 18 \Rightarrow 36 b = 18 \Rightarrow b = \frac{18}{36} = \frac{1}{2}$

That rules out option B

(A) $f$ demonstrates exponential decay because $a > 0$ and $0 < b < 1$

Worked Example

The function $f$ is given by $f (x) = 8.762 \cdot {(1.01)}^{x}$ . Determine the end behavior of $g$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

Answer:

The expression for $f$ is in the general form for an exponential function, $a b^{x}$

with $a > 0$
and $b > 1$

That means the function will increase without bound as $x$ increases without bound

Write that in proper limit notation

$\lim_{x \to \infty} f (x) = \infty$

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General Exponential Functions (College Board AP® Precalculus): Revision Note

General exponential functions

What is the general form of an exponential function?

What is the domain of an exponential function?

What key characteristics does an exponential function have?

How can you identify a 'hidden' exponential function?

What are the end behaviors of an exponential function?

Examiner Tips and Tricks

The natural base e

What is the natural base e?

Worked Example

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis