Graph and define exponential functions

Ontario Grade 11 Mathematics

Exponential Functions

MCR3U – Unit B1.1 | Understanding, Graphing, and Defining Exponential Functions

You already know how to evaluate powers like

2^3 = 8

or

5^{-1} = 0.2

. In previous grades you worked with linear functions (straight lines) and quadratic functions (parabolas). Both of those have the variable in the base — for example,

x^2

. An exponential function flips that idea: the variable sits in the exponent, and the base is a fixed positive number. That one change produces dramatically different behaviour — quantities that double, triple, or shrink by a fixed proportion with every step. Population growth, radioactive decay, and compound interest all follow this pattern. This lesson builds the definition carefully, connects it to a table of values and a graph, and develops the key vocabulary you need for the rest of the exponential unit.

What you will learn

Define an exponential function and identify its key features, including base, exponent, and restrictions on the base.
Evaluate an exponential function for given input values using function notation.
Sketch the graph of an exponential function and describe its key characteristics (domain, range, intercepts, and end behaviour).
Distinguish between exponential growth and exponential decay from an equation and from a graph.
Compare how changing the base of an exponential function affects the shape and steepness of its graph.

Prerequisite Bridge: Powers and Function Notation

Before defining an exponential function, recall two tools from earlier courses. First, integer and rational exponents: for any real base

b

and positive integer

n

, the expression

b^n

means

b

multiplied by itself

n

times, and

b^0 = 1

for any non-zero base. Negative exponents mean reciprocals:

b^{-n} = \frac{1}{b^n}

. These rules still apply when the exponent is a variable.

Second, function notation: writing

f(x)

simply means 'the output of function

f

when the input is

x

'. So

f(3)

means substitute

x = 3

into the rule for

f

and calculate. Nothing about this changes when the rule is exponential — you still substitute and simplify.

$b^0 = 1$ for any non-zero base $b$ .
$b^{-n} = \frac{1}{b^n}$ — a negative exponent means the reciprocal.
$f(x)$ is read 'f of x' and means substitute $x$ into the function rule.
Exponent rules from Grade 10 (product, quotient, power of a power) all still apply.

Defining an Exponential Function

An exponential function is a function of the form

f(x) = b^x

, where

b

is a positive constant called the base, and

x

is the variable (the exponent). The base must satisfy

b > 0

and

b \neq 1

. Understanding why these restrictions exist is important.

Why must

b > 0

? If

b

were negative, powers like

b^{1/2}

(the square root) would not produce real numbers. For example,

(-4)^{1/2} = \sqrt{-4}

, which is not a real number. Allowing negative bases would create gaps in the domain, so we exclude them.

Why must

b \neq 1

? If

b = 1

, then

1^x = 1

for every value of

x

. That gives a constant function — a flat horizontal line — which has none of the interesting growing or shrinking behaviour we associate with exponential functions. It is technically valid but not useful here, so we exclude it by definition.

The domain of

f(x) = b^x

is all real numbers, written \{x ∈

\mathbb{R}

\}, because you can raise a positive base to any power and get a real result. The range is all positive real numbers, \{y ∈

\mathbb{R}

\mid y > 0\}, because a positive base raised to any power is always positive — it never equals zero and never goes negative.

f(x) = b^x, b > 0, b ≠ 1

An exponential function has the form $f(x) = b^x$ with $b > 0$ and $b \neq 1$ .
The base $b$ must be positive so that all real-number inputs produce real outputs.
The base $b \neq 1$ ensures the function actually grows or shrinks rather than staying flat.
Domain: all real numbers. Range: all positive real numbers (the graph never touches the $x$ -axis).
The $y$ -intercept is always $1$ because $b^0 = 1$ for any valid base.

Graphing Exponential Functions: Growth vs. Decay

To graph

f(x) = b^x

, build a table of values for several integer inputs, plot the points, and connect them with a smooth curve. The shape of the curve depends entirely on whether

b > 1

or

0 < b < 1

.

When

b > 1

, the function models exponential growth. As

x

increases,

f(x)

increases rapidly. As

x

decreases (moves left),

f(x)

gets closer and closer to zero but never actually reaches it. The

x

-axis acts as a horizontal asymptote — a line the graph approaches but never crosses. The graph rises steeply to the right and flattens toward the left.

When

0 < b < 1

, the function models exponential decay. As

x

increases,

f(x)

decreases and approaches zero. As

x

decreases,

f(x)

grows without bound. The graph falls steeply to the right and rises to the left. The

x

-axis is still a horizontal asymptote.

A horizontal asymptote is an imaginary horizontal line that the graph gets infinitely close to but never touches. For

f(x) = b^x

, that asymptote is always the

x

-axis, written as

y = 0

. This tells you the range cannot include zero or any negative number.

Comparing bases: for two growth functions

f(x) = 2^x

and

g(x) = 5^x

, both pass through

(0, 1)

and both increase to the right. However,

g

grows much more steeply because a larger base means each unit increase in

x

multiplies the output by a larger factor. For two decay functions

f(x) = (0.8)^x

and

g(x) = (0.3)^x

,

g

decreases more rapidly because the base

0.3

is further from

1

than

0.8

is.

If $b > 1$ : exponential growth — graph rises left to right, approaches the $x$ -axis on the left.
If $0 < b < 1$ : exponential decay — graph falls left to right, approaches the $x$ -axis on the right.
The horizontal asymptote is $y = 0$ for any function of the form $f(x) = b^x$ .
Every graph of $f(x) = b^x$ passes through $(0, 1)$ because $b^0 = 1$ .
A larger base (when $b > 1$ ) produces steeper growth; a base closer to zero (when $0 < b < 1$ ) produces faster decay.

Key Characteristics Summary

When describing the graph of an exponential function, mathematicians focus on five characteristics: domain, range, intercepts, asymptote, and end behaviour. Knowing these five features lets you sketch or interpret any basic exponential graph quickly and precisely.

The domain is always all real numbers because any real number is a valid exponent for a positive base. The range is always

y > 0

because the output is a positive number raised to a power. There is always a

y

-intercept at

(0, 1)

, and there is never an

x

-intercept because the graph never crosses the

x

-axis. The horizontal asymptote is

y = 0

.

End behaviour describes what happens to

f(x)

as

x

becomes very large (written x \to +\infty) or very small (written x \to -\infty

). For growth (

b > 1

): as

x \to +\infty, f(x) \to +\infty; as x \to -\infty, f(x) \to 0

. For decay (

0 < b < 1

): as

x \to +\infty, f(x) \to 0; as x \to -\infty, f(x) \to +\infty.

Domain: \{x ∈ $\mathbb{R}$ \}. Range: \{y ∈ $\mathbb{R}$ \mid y > 0\}.
$y$ -intercept: $(0, 1)$ . No $x$ -intercept.
Horizontal asymptote: $y = 0$ for the base form $f(x) = b^x$ .
Growth ( $b > 1$ ): rises right, approaches zero left.
Decay ( $0 < b < 1$ ): approaches zero right, rises left.

Key Features of f(x) = b^x: Growth vs. Decay at a Glance

Feature	Growth (b > 1)	Decay (0 < b < 1)
Example base	$b = 3$	$b = \frac{1}{3}$
Domain	All real numbers	All real numbers
Range	$y > 0$	$y > 0$
y-intercept	$(0, 1)$	$(0, 1)$
Horizontal asymptote	$y = 0$	$y = 0$
As x \to +\infty	f(x) \to +\infty	f(x) \to 0
As x \to -\infty	f(x) \to 0	f(x) \to +\infty
Graph direction	Rises left to right	Falls left to right

Worked example

Example 1 – Building a Table and Sketching the Graph of an Exponential Growth Function

Let

f(x) = 3^x

. (a) Evaluate

f(x)

for x ∈ \{-2, -1, 0, 1, 2, 3\}. (b) State the domain, range,

y

-intercept, and horizontal asymptote. (c) Describe the end behaviour and identify whether this is growth or decay.

Substitute each input value
Replace $x$ with each integer in the set and apply exponent rules. For negative exponents, use $3^{-n} = \frac{1}{3^n}$ . Calculate each output carefully: $f(-2) = 3^{-2} = \frac{1}{9}$ , $f(-1) = 3^{-1} = \frac{1}{3}$ , $f(0) = 3^{0} = 1$ , $f(1) = 3^{1} = 3$ , $f(2) = 3^{2} = 9$ , $f(3) = 3^{3} = 27$ .
$f(x) = 3^x$
Record the table of values
Organize the six input-output pairs. Notice that each time $x$ increases by $1$ , the output is multiplied by $3$ . This constant multiplication factor is the hallmark of an exponential function and explains its rapid growth.
Identify domain and range
Because any real number can be used as an exponent of $3$ , the domain is all real numbers. The outputs $\frac{1}{9}, \frac{1}{3}, 1, 3, 9, 27$ are all positive, and this holds for every real $x$ — the base $3$ raised to any power can never be zero or negative. So the range is all positive real numbers.
$\text{Domain: } \{x ∈ \mathbb{R}\}, \text{Range: } \{y ∈ \mathbb{R} \mid y > 0\}$
State the y-intercept and asymptote
At $x = 0$ , $f(0) = 3^0 = 1$ , so the $y$ -intercept is the point $(0, 1)$ . As $x$ decreases, $f(x)$ gets smaller and smaller, approaching zero but never reaching it. The line $y = 0$ (the $x$ -axis) is the horizontal asymptote.
$y\text{-intercept: } (0,1), \text{asymptote: } y = 0$
Describe end behaviour and classify
Since the base $3$ is greater than $1$ , this is an exponential growth function. As $x$ increases without limit, $f(x)$ increases without limit. As $x$ decreases without limit, $f(x)$ approaches $0$ . The graph rises steeply to the right and hugs the $x$ -axis to the left.

Answer: Table:

(-2,\, \frac{1}{9})

,

(-1,\, \frac{1}{3})

,

(0,\, 1)

,

(1,\, 3)

,

(2,\, 9)

,

(3,\, 27)

. Domain: all real numbers. Range:

y > 0

.

y

-intercept:

(0, 1)

. Horizontal asymptote:

y = 0

. Exponential growth because

b = 3 > 1

.

Check: Verify a midpoint:

3^2 = 9

✓. Verify a negative exponent:

3^{-2} = \frac{1}{3^2} = \frac{1}{9}

✓. Each successive output is exactly

3

times the previous one:

\frac{1}{9} \times 3 = \frac{1}{3}

,

\frac{1}{3} \times 3 = 1

,

1 \times 3 = 3

✓. All outputs are positive, confirming range

y > 0

✓.

Worked example

Example 2 – Identifying and Comparing Two Exponential Functions From Their Equations

Two functions are defined as

g(x) = \left(\frac{1}{4}\right)^x

and

h(x) = 4^x

. (a) Classify each as growth or decay and justify your answer. (b) Evaluate both functions at

x = -1

,

x = 0

, and

x = 2

. (c) Explain how the graphs of

g

and

h

are related to each other.

Identify the base of each function
For $g(x)$ , the base is $\frac{1}{4}$ . Since $0 < \frac{1}{4} < 1$ , this is exponential decay — outputs decrease as $x$ increases. For $h(x)$ , the base is $4$ . Since $4 > 1$ , this is exponential growth — outputs increase as $x$ increases.
$b_g = \frac{1}{4}, b_h = 4$
Evaluate g(x) at the three inputs
Substitute $x = -1$ : $g(-1) = \left(\frac{1}{4}\right)^{-1}$ . A negative exponent means take the reciprocal, so $\left(\frac{1}{4}\right)^{-1} = 4$ . Substitute $x = 0$ : $g(0) = \left(\frac{1}{4}\right)^{0} = 1$ . Substitute $x = 2$ : $g(2) = \left(\frac{1}{4}\right)^{2} = \frac{1}{16}$ .
$g(-1) = 4, g(0) = 1, g(2) = \frac{1}{16}$
Evaluate h(x) at the three inputs
Substitute $x = -1$ : $h(-1) = 4^{-1} = \frac{1}{4}$ . Substitute $x = 0$ : $h(0) = 4^{0} = 1$ . Substitute $x = 2$ : $h(2) = 4^{2} = 16$ .
$h(-1) = \frac{1}{4}, h(0) = 1, h(2) = 16$
Compare the outputs at each x-value
At $x = -1$ : $g$ gives $4$ and $h$ gives $\frac{1}{4}$ — these are reciprocals. At $x = 0$ : both give $1$ — both graphs share the $y$ -intercept $(0,1)$ . At $x = 2$ : $g$ gives $\frac{1}{16}$ and $h$ gives $16$ — again reciprocals. This pattern holds for all $x$ : $g(x) = \left(\frac{1}{4}\right)^x = \frac{1}{4^x} = 4^{-x} = h(-x)$ .
$g(x) = 4^{-x} = h(-x)$
Describe the graphical relationship
Because $g(x) = h(-x)$ , the graph of $g$ is the reflection of the graph of $h$ across the $y$ -axis. Every point $(a, c)$ on the graph of $h$ corresponds to the point $(-a, c)$ on the graph of $g$ . Both graphs share the point $(0, 1)$ — the only point on the $y$ -axis — and both have the horizontal asymptote $y = 0$ .

Answer:

g(x) = \left(\frac{1}{4}\right)^x

is exponential decay (

b < 1

);

h(x) = 4^x

is exponential growth (

b > 1

). Values:

g(-1)=4

,

g(0)=1

,

g(2)=\frac{1}{16}

;

h(-1)=\frac{1}{4}

,

h(0)=1

,

h(2)=16

. The graph of

g

is the reflection of the graph of

h

in the

y

-axis because

g(x) = h(-x)

.

Check: Check

g(2)

:

\left(\frac{1}{4}\right)^2 = \frac{1^2}{4^2} = \frac{1}{16}

✓. Check that outputs are reciprocals:

4 \times \frac{1}{4} = 1

✓ and

16 \times \frac{1}{16} = 1

✓. Both functions have all positive outputs, confirming range

y > 0

for each ✓.

Common mistakes and how to avoid them

Writing

f(x) = x^b

(variable in the base) and calling it exponential. For example, confusing

f(x) = x^3

with

f(x) = 3^x

.

Correction: In an exponential function, the variable

x

is always the exponent and the base

b

is the fixed constant.

x^3

is a power function (cubic), not an exponential function.

Thinking that

b^0 = 0

instead of

b^0 = 1

, which leads to an incorrect

y

-intercept of

(0, 0)

.

Correction: Any non-zero base raised to the power of zero equals

1

, so the

y

-intercept of

f(x) = b^x

is always

(0, 1)

, not the origin.

Believing an exponential function can produce zero or negative outputs, and therefore drawing the graph crossing or touching the

x

-axis.

Correction: A positive base raised to any real power is always positive. The

x

-axis (

y = 0

) is a horizontal asymptote — the graph gets close but never touches it.

Applying the negative exponent incorrectly: writing

\left(\frac{1}{4}\right)^{-1} = -4

instead of

4

.

Correction: A negative exponent means take the reciprocal of the base:

\left(\frac{1}{4}\right)^{-1} = \frac{1}{(1/4)} = 4

. The reciprocal of a fraction flips numerator and denominator.

Thinking that a base between 0 and 1 means the function values go negative as

x

increases.

Correction: Bases between 0 and 1 produce outputs that approach zero (decay) but never become zero or negative. All outputs remain in the range

y > 0

.

Lesson summary

An exponential function has the form $f(x) = b^x$ where the base $b$ satisfies $b > 0$ and $b \neq 1$ , and the variable $x$ is the exponent.
The domain of any function $f(x) = b^x$ is all real numbers; the range is all positive real numbers ( $y > 0$ ).
The graph always passes through the $y$ -intercept $(0, 1)$ and has a horizontal asymptote at $y = 0$ .
When $b > 1$ , the function models exponential growth — values increase as $x$ increases. When $0 < b < 1$ , the function models exponential decay — values decrease as $x$ increases.
The graph of $f(x) = \left(\frac{1}{b}\right)^x$ is the reflection of the graph of $f(x) = b^x$ across the $y$ -axis, because $\left(\frac{1}{b}\right)^x = b^{-x}$ .

Check your understanding

Question 1

Which of the following is an exponential function?

$f(x) = x^5$
$f(x) = 5^x$
$f(x) = 5x$
$f(x) = x^{1/5}$

Show answer and explanation

f(x) = 5^x

f(x) = 5^x

has the variable in the exponent and a fixed positive base — this matches the definition

f(x) = b^x

with

b = 5 > 1

. The other options all have the variable in the base, making them a power function, linear function, and radical function respectively.

Question 2

What are the domain and range of

f(x) = 7^x

?

Domain: $x > 0$ ; Range: all real numbers
Domain: all real numbers; Range: $y > 0$
Domain: all real numbers; Range: all real numbers
Domain: $x > 0$ ; Range: $y > 0$

Show answer and explanation

Domain: all real numbers; Range:

y > 0

Any real number can be used as an exponent of

7

, so the domain is all real numbers. Because

7^x

is always positive for any real

x

, the range is

y > 0

. The output can never be zero or negative.

Question 3

The graph of

f(x) = \left(\frac{1}{5}\right)^x

has what type of behaviour and what horizontal asymptote?

Exponential growth; asymptote $y = 1$
Exponential decay; asymptote $y = 1$
Exponential growth; asymptote $y = 0$
Exponential decay; asymptote $y = 0$

Show answer and explanation

Exponential decay; asymptote

y = 0

The base

\frac{1}{5}

satisfies

0 < \frac{1}{5} < 1

, so this is exponential decay — the graph falls as

x

increases. The horizontal asymptote for any function of the form

f(x) = b^x

is

y = 0

, not

y = 1

.

Question 4

If

g(x) = 2^x

, what is the value of

g(-3)

?

$-8$
$\frac{1}{8}$
$6$
$8$

Show answer and explanation

\frac{1}{8}

Substitute

x = -3

:

g(-3) = 2^{-3}

. A negative exponent means take the reciprocal:

2^{-3} = \frac{1}{2^3} = \frac{1}{8}

. The answer is

\frac{1}{8}

, a positive value less than

1

— consistent with the range

y > 0

.

Key terms

Exponential function: A function of the form $f(x) = b^x$ where the base $b$ is a positive constant not equal to $1$ , and the variable $x$ is the exponent.
Base: The fixed positive number $b$ in an exponential function that is repeatedly multiplied. It must satisfy $b > 0$ and $b \neq 1$ .
Exponent: The variable or number that tells how many times the base is used as a factor. In $f(x) = b^x$ , the exponent is $x$ .
Horizontal asymptote: A horizontal line that a graph gets infinitely close to but never crosses. For $f(x) = b^x$ , the horizontal asymptote is $y = 0$ .
Exponential growth: The behaviour of $f(x) = b^x$ when $b > 1$ : outputs increase as $x$ increases, and the graph rises from left to right.
Exponential decay: The behaviour of $f(x) = b^x$ when $0 < b < 1$ : outputs decrease toward zero as $x$ increases, and the graph falls from left to right.
Domain: The set of all allowed input values for a function. For $f(x) = b^x$ , the domain is all real numbers.
Range: The set of all possible output values of a function. For $f(x) = b^x$ , the range is all positive real numbers ( $y > 0$ ).

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