Describe domain, range, intercepts, intervals, and asymptotes of exponential functions

Ontario Grade 11 Mathematics

Exponential Functions

MCR3U – Exponential Functions (Expectation B1.4)

You already know how to evaluate powers such as

2^3 = 8

and

2^{-1} = 0.5

from Grade 9 and 10. In this lesson you will study a whole family of functions built on that idea — exponential functions — and learn how to describe their key features precisely. By the end you will be able to read a function rule such as

f(x) = 3 \cdot 2^x

and immediately state its domain, range, intercepts, intervals of increase or decrease, and asymptote. These descriptions are the foundation for comparing, sketching, and applying exponential models throughout the rest of the course.

What you will learn

State the domain and range of an exponential function and explain why they have those values.
Identify the y-intercept of an exponential function and explain why a horizontal asymptote prevents an x-intercept.
Describe the intervals on which an exponential function is positive, increasing, or decreasing.
Identify the horizontal asymptote of an exponential function and explain what it means graphically.
Apply these five descriptors to exponential functions of the form $f(x) = a \cdot b^x$ where $b > 0$ , $b \neq 1$ .

Prerequisite Bridge: Powers With Any Exponent

Before looking at exponential functions, it helps to recall a few power rules from Grade 10. When the base is a positive number, you can raise it to any real-number exponent and always get a positive result. For example,

2^0 = 1

,

2^{0.5} \approx 1.41

, and

2^{-3} = 0.125

. The output is never zero and never negative.

This single fact — a positive base raised to any real exponent always gives a positive output — explains most of the key features you will describe in this lesson. Keep it in mind as you read each section.

A positive base raised to any real exponent always produces a positive result.
$b^0 = 1$ for any base $b \neq 0$ .
Negative exponents give fractions, not negative numbers: $b^{-n} = \frac{1}{b^n}$ .

What Is an Exponential Function?

An exponential function has the form

f(x) = a \cdot b^x

, where

a

is a non-zero constant called the vertical stretch factor,

b

is a positive constant called the base, and

b \neq 1

. The variable

x

sits in the exponent — that is what makes it exponential, not polynomial.

The base

b

controls the direction of the function. When

b > 1

, the function grows larger as

x

increases — this is called exponential growth. When

0 < b < 1

, the function shrinks toward zero as

x

increases — this is called exponential decay. Both types share the same five key features you are about to learn.

The simplest example is

f(x) = 2^x

, which has

a = 1

and

b = 2

. A decay example is

g(x) = \left(\frac{1}{2}\right)^x

, which has

a = 1

and

b = \frac{1}{2}

. You will use both types when practising the five descriptors below.

f(x) = a · b^x

Standard form: $f(x) = a \cdot b^x$ , with $a \neq 0$ , $b > 0$ , $b \neq 1$ .
$b > 1$ gives exponential growth; $0 < b < 1$ gives exponential decay.
The variable is in the exponent, not the base.

Domain and Range

The domain of a function is the set of all permitted input values (

x

-values). For

f(x) = a \cdot b^x

, you can substitute any real number for

x

— positive, negative, zero, fractional — and the expression is always defined. Therefore the domain is all real numbers, written in set notation as \{x ∈

\mathbb{R}

\} or in interval notation as

(-\infty, \infty)

.

The range is the set of all possible output values (

y

-values). Because a positive base raised to any power is always positive, the factor

b^x

is always greater than zero. If

a > 0

, then

a \cdot b^x

is always positive, so every output is above zero. If

a < 0

, every output is below zero. In the most common case where

a > 0

, the range is \{y ∈

\mathbb{R}

\mid y > 0\}, or in interval notation

(0, \infty)

. The function never actually reaches zero — it only approaches it.

This is different from a linear or quadratic function, whose outputs can pass through zero and become negative. The range being strictly positive (when

a > 0

) is a defining characteristic of exponential functions.

Domain: all real numbers, $(-\infty, \infty)$ .
Range (when $a > 0$ ): all positive real numbers, $(0, \infty)$ .
Range (when $a < 0$ ): all negative real numbers, $(-\infty, 0)$ .
The output never equals zero for any finite value of $x$ .

Intercepts and the Horizontal Asymptote

The y-intercept is the output when

x = 0

. Substituting

x = 0

into

f(x) = a \cdot b^x

gives

f(0) = a \cdot b^0 = a \cdot 1 = a

. So the y-intercept is always equal to

a

, the vertical stretch factor. For

f(x) = 3 \cdot 2^x

, the y-intercept is

(0, 3)

.

An x-intercept would occur where the output equals zero. But, as established above,

a \cdot b^x

is never zero when

a \neq 0

and

b > 0

. Therefore exponential functions of this form have no x-intercept. The graph never crosses the x-axis.

This connects directly to the horizontal asymptote. An asymptote is a line that the graph approaches but never actually reaches. For

f(x) = a \cdot b^x

, as

x

decreases without bound (moves far to the left when

b > 1

, or far to the right when

0 < b < 1

), the output gets closer and closer to zero but never touches it. The line

y = 0

(the x-axis) is therefore a horizontal asymptote. The graph hugs this line from above (if

a > 0

) or from below (if

a < 0

) but never crosses it.

To summarise: y-intercept at

(0, a)

, no x-intercept, and horizontal asymptote at

y = 0

.

f(0) = a · b^0 = a

y-intercept: $(0, a)$ — found by setting $x = 0$ .
No x-intercept: $a \cdot b^x = 0$ has no solution.
Horizontal asymptote: the line $y = 0$ .
The graph approaches the asymptote but never reaches it.

Intervals of Increase and Decrease

An interval of increase is a portion of the domain where the output values rise as

x

moves to the right. An interval of decrease is a portion where outputs fall as

x

moves right.

For a growth function (

b > 1

,

a > 0

): as

x

increases,

b^x

increases, so

f(x)

increases over its entire domain. The function is increasing on

(-\infty, \infty)

. A classic example is

f(x) = 2^x

: at

x = 0

the value is

1

, at

x = 1

it is

2

, at

x = 2

it is

4

— always rising.

For a decay function (

0 < b < 1

,

a > 0

): as

x

increases,

b^x

decreases (think of repeated multiplication by a fraction less than one), so

f(x)

decreases over its entire domain. The function is decreasing on

(-\infty, \infty)

. For example,

g(x) = \left(\frac{1}{2}\right)^x

: at

x = 0

the value is

1

, at

x = 1

it is

0.5

, at

x = 2

it is

0.25

— always falling.

An important point: an exponential function with

a > 0

is either always increasing or always decreasing — it never changes direction. It has no maximum or minimum turning point.

When $b > 1$ and $a > 0$ : increasing on $(-\infty, \infty)$ .
When $0 < b < 1$ and $a > 0$ : decreasing on $(-\infty, \infty)$ .
An exponential function never changes direction — no peaks or valleys.
When $a < 0$ , the inequality directions reverse.

Summary of Key Features for $f(x) = a \cdot b^x$

Feature	Growth: $b > 1$ , $a > 0$	Decay: $0 < b < 1$ , $a > 0$	Decay base, $a < 0$
Domain	All real numbers	All real numbers	All real numbers
Range	$(0, \infty)$	$(0, \infty)$	$(-\infty, 0)$
y-intercept	$(0, a)$	$(0, a)$	$(0, a)$
x-intercept	None	None	None
Asymptote	$y = 0$	$y = 0$	$y = 0$
Interval	Increasing on $(-\infty, \infty)$	Decreasing on $(-\infty, \infty)$	Increasing on $(-\infty, \infty)$

Worked example

Describing a Growth Function

For the function

f(x) = 4 \cdot 3^x

, state the domain, range, y-intercept, x-intercept (if any), horizontal asymptote, and whether the function is increasing or decreasing. Justify each answer.

Identify the parameters
Write the function in standard form $f(x) = a \cdot b^x$ and read off $a$ and $b$ . Here $a = 4$ and $b = 3$ . Since $b = 3 > 1$ , this is an exponential growth function. Since $a = 4 > 0$ , all outputs will be positive.
$a = 4, b = 3$
State the domain
Any real number can be substituted for $x$ in $3^x$ without causing an error, so the domain is all real numbers.
$\text{Domain: } (-\infty, \infty)$
State the range
Because $a = 4 > 0$ and $3^x > 0$ for all $x$ , the product $4 \cdot 3^x$ is always strictly positive. The output can get arbitrarily large but never reaches zero.
$\text{Range: } (0, \infty)$
Find the y-intercept
Substitute $x = 0$ into the function. Use the rule $b^0 = 1$ to simplify.
$f(0) = 4 · 3^0 = 4 · 1 = 4$
Check for an x-intercept
An x-intercept requires $f(x) = 0$ . Since $4 \cdot 3^x$ is always positive, it can never equal zero. There is no x-intercept.
$4 · 3^x > 0 \text{ for all } x$
Identify the horizontal asymptote
As $x$ decreases without bound, $3^x$ gets closer and closer to zero, so $4 \cdot 3^x$ approaches zero from above but never reaches it. The horizontal asymptote is the line $y = 0$ .
$y = 0$
Determine increase or decrease
Since $b = 3 > 1$ and $a = 4 > 0$ , larger values of $x$ produce larger outputs. The function is increasing on its entire domain.
$\text{Increasing on } (-\infty, \infty)$

Answer: Domain:

(-\infty, \infty)

. Range:

(0, \infty)

. y-intercept:

(0, 4)

. No x-intercept. Horizontal asymptote:

y = 0

. Increasing on

(-\infty, \infty)

.

Check: At

x = 1

:

f(1) = 4 \cdot 3 = 12 > 4 = f(0)

. At

x = -1

:

f(-1) = 4 \cdot 3^{-1} = \frac{4}{3} \approx 1.33 < 4 = f(0)

. The outputs rise as

x

rises, confirming the function is increasing. Both outputs are positive, consistent with the stated range.

Worked example

Describing a Decay Function With a Negative Stretch Factor

For the function

h(x) = -5 \cdot \left(\frac{1}{4}\right)^x

, state the domain, range, y-intercept, x-intercept (if any), horizontal asymptote, and whether the function is increasing or decreasing. Justify each answer.

Identify the parameters
Compare with $f(x) = a \cdot b^x$ . Here $a = -5$ and $b = \frac{1}{4}$ . Since $0 < b = \frac{1}{4} < 1$ , this is a decay-type base. Since $a = -5 < 0$ , all outputs will be negative — the graph sits entirely below the x-axis.
$a = -5, b = \frac{1}{4}$
State the domain
There is no restriction on $x$ ; any real number is a valid input for $\left(\frac{1}{4}\right)^x$ .
$\text{Domain: } (-\infty, \infty)$
State the range
The factor $\left(\frac{1}{4}\right)^x$ is always strictly positive. Multiplying by $a = -5$ flips every output to a strictly negative value. The output approaches zero but never reaches it, so the range is all negative real numbers.
$\text{Range: } (-\infty, 0)$
Find the y-intercept
Set $x = 0$ and evaluate. Recall $\left(\frac{1}{4}\right)^0 = 1$ .
$h(0) = -5 · (\frac{1}{4})^0 = -5 · 1 = -5$
Check for an x-intercept
An x-intercept requires $h(x) = 0$ . Since $-5 \cdot \left(\frac{1}{4}\right)^x$ is always strictly negative, it never equals zero. There is no x-intercept.
$-5 · (\frac{1}{4})^x < 0 \text{ for all } x$
Identify the horizontal asymptote
As $x$ increases without bound, $\left(\frac{1}{4}\right)^x$ gets arbitrarily close to zero, so $h(x)$ approaches zero from below. As $x$ decreases without bound, the outputs become very large in magnitude (very negative) and move away from zero. The horizontal asymptote is still $y = 0$ , approached from below.
$y = 0$
Determine increase or decrease
As $x$ increases, $\left(\frac{1}{4}\right)^x$ decreases toward zero (positive and shrinking), and multiplying by $-5$ means the output increases toward zero from below. So $h(x)$ is increasing on its entire domain — the outputs move upward (less negative) as $x$ grows.
$\text{Increasing on } (-\infty, \infty)$

Answer: Domain:

(-\infty, \infty)

. Range:

(-\infty, 0)

. y-intercept:

(0, -5)

. No x-intercept. Horizontal asymptote:

y = 0

. Increasing on

(-\infty, \infty)

.

Check: At

x = 1

:

h(1) = -5 \cdot \frac{1}{4} = -1.25

. At

x = 0

:

h(0) = -5

. At

x = -1

:

h(-1) = -5 \cdot 4 = -20

. As

x

goes from

-1

to

0

to

1

, the outputs go from

-20

to

-5

to

-1.25

: rising (increasing). All outputs are negative, confirming the range

(-\infty, 0)

.

Common mistakes and how to avoid them

Saying the range is all real numbers, including zero and negative values, when

a > 0

.

Correction: The output of

a \cdot b^x

is always strictly positive when

a > 0

and

b > 0

. The range is

(0, \infty)

, never including zero.

Claiming the x-intercept is

(0, 0)

or that the graph eventually crosses the x-axis.

Correction: The x-axis is the horizontal asymptote. The graph approaches it but never touches or crosses it. There is no x-intercept.

Confusing the y-intercept with the base: writing

b

instead of

a

as the y-intercept.

Correction: Substituting

x = 0

gives

a \cdot b^0 = a \cdot 1 = a

. The y-intercept is

a

, not

b

.

Concluding that a decay function (

0 < b < 1

) with a negative

a

is decreasing because it decays.

Correction: When

a < 0

and

0 < b < 1

, the outputs are negative and become less negative as

x

increases, so the function is actually increasing. Check by computing two outputs.

Stating the domain as only positive real numbers because the outputs are positive.

Correction: Positive outputs describe the range, not the domain. The input

x

can be any real number, including negatives and fractions.

Lesson summary

An exponential function has the form $f(x) = a \cdot b^x$ with $a \neq 0$ , $b > 0$ , and $b \neq 1$ . The variable is in the exponent.
The domain is always all real numbers: $(-\infty, \infty)$ .
The range is $(0, \infty)$ when $a > 0$ , and $(-\infty, 0)$ when $a < 0$ — zero is never included.
The y-intercept is always $(0, a)$ . There is never an x-intercept because the output cannot equal zero.
The horizontal asymptote is always the line $y = 0$ , approached but never reached.
The function is increasing on all of $(-\infty, \infty)$ when $b > 1$ and $a > 0$ (or $0 < b < 1$ and $a < 0$ ), and decreasing when $0 < b < 1$ and $a > 0$ (or $b > 1$ and $a < 0$ ).

Check your understanding

Question 1

What is the y-intercept of

f(x) = 7 \cdot 5^x

?

$(0, 5)$
$(0, 7)$
$(0, 35)$
$(0, 1)$

Show answer and explanation

(0, 7)

Substitute

x = 0

:

f(0) = 7 \cdot 5^0 = 7 \cdot 1 = 7

. The y-intercept is

(0, 7)

, which equals

a

, not

b

.

Question 2

Which statement correctly describes the range of

g(x) = 2 \cdot \left(\frac{1}{3}\right)^x

?

All real numbers
All real numbers greater than or equal to zero
All real numbers greater than zero
All real numbers less than zero

Show answer and explanation

All real numbers greater than zero

Since

a = 2 > 0

and

\left(\frac{1}{3}\right)^x > 0

for all

x

, the product is always strictly positive. The range is

(0, \infty)

— zero is not included because the graph only approaches

y = 0

as an asymptote.

Question 3

The graph of

f(x) = 6 \cdot 4^x

has a horizontal asymptote. Which line is it, and why?

$y = 6$ , because $a = 6$
$y = 4$ , because $b = 4$
$y = 0$ , because the output approaches zero for very negative $x$ -values
$y = 1$ , because $b^0 = 1$

Show answer and explanation

y = 0

, because the output approaches zero for very negative

x

-values

As

x

decreases without bound,

4^x

approaches zero, so

6 \cdot 4^x

approaches zero from above. The horizontal asymptote is

y = 0

. The values

a = 6

and

b = 4

are parameters, not asymptotes.

Question 4

On which interval is

p(x) = 3 \cdot \left(\frac{2}{5}\right)^x

decreasing?

Only for $x > 0$
Only for $x < 0$
$(-\infty, \infty)$
The function is not decreasing anywhere

Show answer and explanation

(-\infty, \infty)

Since

0 < b = \frac{2}{5} < 1

and

a = 3 > 0

, larger

x

-values produce smaller outputs. The function decreases over its entire domain, which is all real numbers:

(-\infty, \infty)

.

Key terms

Exponential function: A function of the form $f(x) = a \cdot b^x$ , where the variable $x$ appears as an exponent, $a \neq 0$ , $b > 0$ , and $b \neq 1$ .
Domain: The complete set of all valid input values ( $x$ -values) for a function.
Range: The complete set of all possible output values ( $y$ -values) produced by a function.
y-intercept: The point where a graph crosses the y-axis, found by substituting $x = 0$ into the function.
x-intercept: The point where a graph crosses the x-axis, found by setting the output equal to zero. Exponential functions of the form $a \cdot b^x$ have none.
Asymptote: A line that a graph approaches as the input or output increases or decreases without bound, but never actually reaches.
Interval of increase: A portion of the domain where output values rise as the input increases (moving left to right on a graph).
Interval of decrease: A portion of the domain where output values fall as the input increases (moving left to right on a graph).

Continue through MCR3U

View the complete MCR3U Ontario Grade 11 Mathematics curriculum and lessons

Tutor support for B1.4 · Describe domain, range, intercepts, intervals, and asymptotes of exponential functions

Working on B1.4 · Describe domain, range, intercepts, intervals, and asymptotes of exponential functions in MCR3U? Bring the question and your own attempt. In a live session, you can label the axes and connect each feature of the graph to the equation or situation. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث B1.4 · توصیف دامنه، برد، عرض‌ها، بازه‌ها و مجانب‌های تابع نمایی در درس MCR3U، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید محورها را برچسب بزنید و هر ویژگی نمودار را به معادله یا موقعیت مسئله ربط دهید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation B1.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question