B. Exponential Functions · Exponential Functions

Key Properties of Exponential Functions

B1.4About 70 minutes

Learning goals

I can state the domain, range, intercepts and asymptote of f(x) = aˣ.
I can tell from the base whether an exponential function is increasing or decreasing.
I can compare exponential functions with different bases.
I can find these properties from a table, mapping diagram, function machine, graph or equation.

Words to know

domain
The set of all allowed input values. For f(x) = aˣ it is all real numbers: {x ∈ ℝ}.

range
The set of all output values. For f(x) = aˣ it is all positive real numbers: {y ∈ ℝ | y > 0}.

horizontal asymptote
A horizontal line that a graph gets closer and closer to (without reaching) as x becomes very large or very small. For y = aˣ it is y = 0.

increasing function
As x increases, y increases. The graph rises from left to right.

decreasing function
As x increases, y decreases. The graph falls from left to right.

intercept
A point where a graph crosses an axis. y = aˣ has a y-intercept of 1 and no x-intercept.

What every exponential function has in common

You have now graphed y = 2ˣ, y = 3ˣ, y = 4ˣ, y = 8ˣ and y = (1/2)ˣ. They look different, but they share a set of key properties. Knowing these properties lets you sketch any exponential graph quickly and check a calculator graph for mistakes. Today you will investigate them using graphs, tables, mapping diagrams and function machines.

Investigation: three bases on one set of axes

Graph f(x) = 2ˣ, g(x) = 3ˣ and h(x) = 0.5ˣ on the same axes (by hand from tables, or with Desmos or a graphing calculator). Then answer:

Where does each graph cross the y-axis? Why?
Does any graph cross the x-axis?
Which graphs go up from left to right and which go down?
For x > 0, which graph is highest? For x < 0?

y = 2ˣ (solid), y = 3ˣ (long dashes) and y = 0.5ˣ (dotted). All three share the y-intercept 1 because a⁰ = 1. None of them reaches the x-axis.

What the investigation shows

y-intercept: all three pass through (0, 1), because f(0) = a⁰ = 1 for every base.
No x-intercept: aˣ is never 0 (it is always positive), so the graph never reaches the x-axis.
Increasing or decreasing: if a > 1 the function increases over its whole domain; if 0 < a < 1 it decreases over its whole domain. It never turns around.
Comparing bases: for x > 0, the bigger base gives the bigger value (3ˣ > 2ˣ). For x < 0 it is the opposite (3⁻¹ = 1/3 is less than 2⁻¹ = 1/2), so y = 3ˣ hugs the x-axis more tightly on the left.

The horizontal asymptote

For y = 2ˣ, as x goes far to the left (x → −∞), y = 1/2, 1/4, 1/8, 1/16, … gets closer and closer to 0 but never reaches it. The line y = 0 (the x-axis) is a horizontal asymptote. For y = 0.5ˣ the same thing happens on the right side (as x → +∞).

Key features of y = 2ˣ: y-intercept 1, no x-intercept, increasing for all x, and the horizontal asymptote y = 0 (the x-axis itself).

Property	f(x) = aˣ, a > 1	f(x) = aˣ, 0 < a < 1
Domain	{x ∈ ℝ} (all real numbers)	{x ∈ ℝ}
Range	{y ∈ ℝ \| y > 0}	{y ∈ ℝ \| y > 0}
y-intercept	1	1
x-intercept	none	none
Horizontal asymptote	y = 0, approached as x → −∞	y = 0, approached as x → +∞
Increasing / decreasing	increasing on its whole domain	decreasing on its whole domain
Example	y = 2ˣ: growth	y = (1/2)ˣ: decay

Key properties of the exponential function f(x) = aˣ.

Other representations

The same properties show up in every representation:

Table of values (equal steps in x): the y-values are multiplied by the same number (the base) each step, and they are all positive.
Mapping diagram: each input has exactly one arrow; no arrow ever lands on 0 or a negative number.
Function machine: the machine "raises a to the input". Whatever goes in, something positive comes out.

A function machine for f(x) = 3ˣ. Inputs −1, 0 and 2 give outputs 1/3, 1 and 9. Any real number can go in (domain ℝ), but only positive numbers come out (range y > 0).

Worked example

Example 1: Stating the properties

State the domain, range, intercepts, interval of increase or decrease, and the equation of the asymptote for f(x) = 5ˣ.

Domain: any x works, so {x ∈ ℝ}.
Range: 5ˣ is always positive and takes every positive value, so {y ∈ ℝ | y > 0}.
y-intercept: f(0) = 5⁰ = 1. x-intercept: none, since 5ˣ is never 0.
The base 5 > 1, so f is increasing for all x. Horizontal asymptote: y = 0, approached as x → −∞.

Domain ℝ; range y > 0; y-intercept 1; no x-intercept; increasing on ℝ; asymptote y = 0.

Worked example

Example 2: A decreasing exponential function

For g(x) = (2/3)ˣ, find g(−2), g(0) and g(3), then describe the graph.

g(−2) = (3/2)² = 9/4. g(0) = 1. g(3) = 8/27.
The base 2/3 is between 0 and 1, so g is decreasing over its whole domain.
As x → +∞, the values get closer to 0: the asymptote is y = 0, approached on the right.
Domain {x ∈ ℝ}, range {y ∈ ℝ | y > 0}.

The graph falls from left to right through (−2, 9/4), (0, 1) and (3, 8/27), approaching the x-axis on the right.

Worked example

Example 3: Properties from a table

A function has this table: x = −2, −1, 0, 1, 2 and y = 16, 4, 1, 1/4, 1/16. Find the equation and describe the properties.

Ratios: 4 ÷ 16 = 1/4, 1 ÷ 4 = 1/4, (1/4) ÷ 1 = 1/4. Each step multiplies y by 1/4.
At x = 0, y = 1, so there is no extra factor in front: f(x) = (1/4)ˣ.
Base between 0 and 1: decreasing. y-intercept 1, no x-intercept, asymptote y = 0, domain ℝ, range y > 0.

f(x) = (1/4)ˣ, a decreasing exponential function.

Worked example

Example 4: Comparing two bases

Without graphing, decide which is greater: (a) 3^2.5 or 2^2.5 (b) 3^(−2) or 2^(−2). Explain using the graphs.

(a) For x > 0 the larger base gives the larger value, so 3^2.5 > 2^2.5 (about 15.6 and 5.7).
(b) For x < 0 the order flips: 3⁻² = 1/9 and 2⁻² = 1/4, so 2⁻² > 3⁻².
On the graphs, y = 3ˣ is above y = 2ˣ to the right of the y-axis and below it to the left. They cross only at (0, 1).

3^2.5 is greater; 2^(−2) is greater.

Just for fun

An asymptote and an exponential curve go on a date. The curve keeps getting closer... and closer... and closer.

They never actually touch. Their friends say they are taking things slowly. Mathematicians say they are taking things infinitely slowly.

Hands-on activity

Guess my base: a function-machine game

You need: a calculator, paper, a pencil and a partner (a family member is perfect).

Your partner secretly chooses a base a (for example 2, 5, 0.5 or 1.5) and writes it on a hidden slip.
You call out inputs. Your partner enters a ^ (your input) on the calculator and tells you only the output. Try x = 0 first, then x = 1, x = 2 and x = −1.
Record the input-output pairs in a table. Find the base. Then decide: is the function increasing or decreasing? What is the y-intercept?
Try to get the machine to output 0 or a negative number. Can you?
Switch roles.

What to look for: an input of 0 always gives 1 (y-intercept), so that first question gives nothing away. An input of 1 gives the base directly. Ratios between outputs for x = 0, 1, 2 are all equal to the base. No input ever produces 0 or a negative number: the range is y > 0.

Your turn

State the domain and range of f(x) = 7ˣ.

What is the y-intercept of y = (0.2)ˣ? Does it have an x-intercept?

Is y = 1.05ˣ increasing or decreasing? Explain.

Give the equation of the horizontal asymptote of y = 4ˣ. On which side of the graph is it approached?

Which is greater, 5⁻¹ or 4⁻¹? Explain using the graphs of y = 5ˣ and y = 4ˣ.

A table has x = 0, 1, 2, 3 and y = 1, 6, 36, 216. Write the function and state whether it is increasing or decreasing.

Explain why 0 is not in the range of f(x) = aˣ.

More practice: warm-up

What is f(0) for f(x) = 12ˣ?

Which base gives a decreasing function: 1.2 or 0.8?

True or false: the graph of y = 3ˣ crosses the x-axis at x = −3.

Write the range of y = 2ˣ in set notation.

For x > 0, which is higher: y = 10ˣ or y = 2ˣ?

More practice: on your own

Describe how the graphs of y = 4ˣ and y = (1/4)ˣ are alike and different.

For f(x) = 2ˣ, find the x-value where f(x) = 1/32. What does this tell you about the left side of the graph?

A mapping diagram sends −1 → 10, 0 → 1, 1 → 0.1, 2 → 0.01. Write the function and state its key properties.

Order from least to greatest without a calculator: 2^0.5, 3^0.5, 0.5^0.5.

Can a function f(x) = aˣ ever have f(3) = −8? Explain.

Challenge

For which values of x is 3ˣ < 2ˣ? Explain.

The function f(x) = aˣ passes through (2, 0.36). Find a, then state whether f is increasing or decreasing and on which side the asymptote is approached.

Explain why every exponential function f(x) = aˣ has an inverse that is also a function, using the fact that it is always increasing or always decreasing.

Check your understanding

Quiz

What is the range of f(x) = 6ˣ?

What is the horizontal asymptote of y = (1/3)ˣ?

Which function is decreasing?

Which point lies on both y = 2ˣ and y = 7ˣ?

How many x-intercepts does y = 5ˣ have?

For x < 0, which graph is highest?

For parents and teachers

Big idea: Expectation B1.4: investigate and describe key properties of exponential functions (domain, range, intercepts, increasing/decreasing intervals, asymptotes) from tables, mapping diagrams, graphs, equations and function machines.

How to help: Ask your teen to sketch y = 2ˣ from memory and point to the intercept and the asymptote. Then ask "what changes if the base is 1/2?" Play the guessing game from the activity.

Common difficulties:

Writing the range as y ≥ 0. The value 0 is never reached.
Saying the graph "touches" the x-axis far to the left. On a calculator screen it looks like it does; zoom in to see the gap.
Thinking a larger base always gives a larger value. That is only true for x > 0.
Set notation: {y ∈ ℝ | y > 0} is read "the set of real numbers y such that y is greater than 0".