D. Characteristics of Functions · Combining Functions and Modelling

Quotients of Functions

D2.1About 70 minutes

Learning goals

I can find the quotient f/g of two functions and state its domain, excluding every x where g(x) = 0.
I can predict zeros, vertical asymptotes and holes of a quotient from the zeros of f and g.
I can find the range, maximum and minimum values and number of zeros of a quotient.
I can investigate quotients that mix function families, such as y = (sin x)/x.

Words to know

quotient of functions
The function (f/g)(x) = f(x)/g(x), defined only where g(x) ≠ 0.

vertical asymptote
A vertical line x = a that the graph approaches as the outputs grow without bound. A quotient has one where g(a) = 0 and f(a) ≠ 0.

hole
A single missing point on a graph. It appears when a factor cancels from the top and bottom, so f(a) = 0 and g(a) = 0.

horizontal asymptote
A horizontal line that the graph approaches as x → ∞ or x → −∞.

average cost
Total cost divided by the number of items or people: a quotient C(n)/n.

Splitting the cottage bill

A group rents a cottage on a lake for a summer weekend. The rental is $1200, plus $15 per person for linens. The total cost for n people is C(n) = 1200 + 15n. The cost per person is

A(n) = C(n)/n = (1200 + 15n)/n.

With 4 people, each pays 1260/4 = $315. With 20 people, each pays 1500/20 = $75. As n grows, the $1200 is shared more and more thinly, and A(n) gets closer and closer to $15, but never reaches it. That is a horizontal asymptote, and A is a quotient of two functions.

Remember the sibling unit Rational Functions and Equations, where you graphed reciprocal functions like 1/(x − 3) and quotients of linear functions. This lesson extends the same ideas to quotients of any two functions.

Dividing functions

The quotient of f and g is

(f/g)(x) = f(x)/g(x).

Its domain is the intersection of the domains of f and g, with every x where g(x) = 0 removed.

How to read the graph of f/g from the graphs of f and g:

Zeros: where f(x) = 0 and g(x) ≠ 0.
Vertical asymptote: usually where g(x) = 0 and f(x) ≠ 0. Near there, you divide by a tiny number, so the output is huge.
Hole: where f(x) = 0 and g(x) = 0 because of a common factor that cancels.
Where g(x) = 1, the quotient equals f(x). Where g(x) = −1, it equals −f(x).
Where |g(x)| is large, the quotient is small. Where |g(x)| is small (but not 0), the quotient is large.
Signs: positive where f and g have the same sign, negative where the signs differ.

(x + 1)/(x − 2) (solid) has a vertical asymptote where the denominator g is 0 and a zero where the numerator f is 0. Far left and right it approaches y = 1, the ratio of the leading coefficients.

Worked example

Example 1: A quotient of two lines

Let f(x) = x + 1 and g(x) = x − 2. Find the key features of (f/g)(x).

(f/g)(x) = (x + 1)/(x − 2). Domain: x ≠ 2 (g(2) = 0).
Zero: f(x) = 0 at x = −1, and g(−1) = −3 ≠ 0, so the zero is x = −1. y-intercept: (0 + 1)/(0 − 2) = −1/2.
Vertical asymptote: g(2) = 0 and f(2) = 3 ≠ 0, so x = 2 is a vertical asymptote.
Horizontal asymptote: for very large |x|, (x + 1)/(x − 2) is close to x/x = 1. So y = 1.
Range: the value 1 is never reached, because (x + 1)/(x − 2) = 1 would mean x + 1 = x − 2, which is impossible. Range: y ≠ 1.
Signs: x < −1: (−)/(−) > 0. −1 < x < 2: (+)/(−) < 0. x > 2: (+)/(+) > 0.

Domain x ≠ 2, range y ≠ 1, one zero (x = −1), y-intercept −1/2, asymptotes x = 2 and y = 1, no maximum or minimum.

Worked example

Example 2: A hole and a bump

Find the domain, range and key features of (a) h(x) = (x² − 4)/(x − 2) and (b) k(x) = 1/(x² + 1).

(a) The denominator is 0 at x = 2, so the domain is x ≠ 2.
Factor: (x² − 4)/(x − 2) = (x − 2)(x + 2)/(x − 2) = x + 2, for x ≠ 2. The factor (x − 2) cancels, so there is a hole, not an asymptote, at x = 2. The missing point is (2, 2 + 2) = (2, 4).
The graph is the line y = x + 2 with the point (2, 4) removed. Range: y ≠ 4. One zero: x = −2.
(b) The denominator x² + 1 is at least 1, never 0. So the domain is all real numbers and there is no vertical asymptote.
The denominator is smallest (1) at x = 0, so k is largest there: maximum point (0, 1).
As |x| grows, x² + 1 grows, so k(x) → 0: horizontal asymptote y = 0. Since k(x) > 0, there are no zeros. Range: 0 < y ≤ 1.

(a) the line y = x + 2 with a hole at (2, 4); (b) a bell-shaped curve with maximum (0, 1), asymptote y = 0, range 0 < y ≤ 1.

Left: when a factor cancels, the quotient is a line with one missing point (an open circle). Right: the denominator x² + 1 (dashed) is never 0, so 1/(x² + 1) (solid) has no vertical asymptote. It is largest where the denominator is smallest.

Quotients that mix families

You already know one famous quotient: tan x = sin x / cos x (from the sibling unit Radian Measure and Trigonometric Functions). Its zeros are the zeros of sin x (x = 0, ±π, ...), and its vertical asymptotes are where cos x = 0 (x = ±π/2, ±3π/2, ...). Exactly the rules above.

Now look at y = (sin x)/x, with x in radians.

Domain: x ≠ 0.
Zeros: where sin x = 0 but x ≠ 0, so x = ±π, ±2π, ±3π, ...
Envelope: since −1 ≤ sin x ≤ 1, the graph stays between y = 1/x and y = −1/x for x > 0 (and between the same two curves, switched, for x < 0). The wiggles shrink as |x| grows.
Near x = 0, sin x and x are almost equal, so the quotient is close to 1 (see the table). Both sin x and x are 0 at x = 0, so the graph has a hole at (0, 1), not an asymptote.

x (radians)	1	0.5	0.1	0.01	−0.01	−0.1
(sin x)/x	0.84147	0.95885	0.99833	0.99998	0.99998	0.99833

As x gets closer to 0 from either side, (sin x)/x gets closer to 1. Values rounded to 5 decimal places.

y = (sin x)/x (solid) has zeros at every nonzero multiple of π. It is squeezed between y = 1/x and y = −1/x (dashed), and it has a hole at (0, 1).

Radians matter here

Try (sin x)/x with your calculator in degree mode and x = 0.01: you get about 0.01745, not 1. That number is π/180. The limit 1 is one of the reasons radians are the natural unit for calculus, which you will meet in Calculus and Vectors (MCV4U).

Worked example

Example 3: Key features of (sin x)/x

For q(x) = (sin x)/x on −3π ≤ x ≤ 3π, x ≠ 0, state the number of zeros, the behaviour near x = 0, and whether q has a maximum value.

Zeros: x = ±π, ±2π, ±3π: six zeros.
Near x = 0, q(x) gets as close to 1 as we like, but q(0) is undefined: a hole at (0, 1).
For x ≠ 0, |sin x| < |x|, so |q(x)| < 1. The values come arbitrarily close to 1 but never reach it.
So q has no maximum value (there is no largest output below 1). Its range on this interval is about −0.217 ≤ y < 1; the lowest points, about (±4.49, −0.217), can be found with graphing technology.

Six zeros, a hole at (0, 1), no maximum value, minimum about −0.217.

Just for fun

Funny but true: ask almost any calculator to divide by zero and it will refuse, showing "Error" or "Math ERROR". No amount of polite button pressing changes its mind. On a graph, that refusal is a vertical asymptote or a hole.

Hands-on activity

Investigate a quotient with a calculator

You need: a scientific calculator, paper and pencil.

Part A (radian mode). Make a table of (sin x)/x for x = 2, 1, 0.5, 0.1, 0.01, and then the same negative values. What number do the outputs approach? Now try x = 0. What does the calculator say?

Part B (degree mode). Repeat for x = 1, 0.1, 0.01. What number do the outputs approach now? Multiply it by 180. What do you get?

Part C (the cottage). Make a table of A(n) = (1200 + 15n)/n for n = 1, 2, 4, 8, 12, 20, 40, 100. Plot it.

What to look for: Part A approaches 1, and x = 0 gives an error. Part B approaches about 0.017453, and 0.017453 × 180 ≈ 3.1415, which is π (so the degree-mode limit is π/180). Part C: 1215, 615, 315, 165, 115, 75, 45, 27. The values fall quickly, then level off toward 15: a horizontal asymptote at A = 15.

Your turn

f(x) = x − 4 and g(x) = x + 3. State the domain, zero, vertical asymptote and horizontal asymptote of f/g.

f(x) = x² − 9 and g(x) = x + 3. Describe the graph of f/g.

Find the domain of y = log x / (x − 5).

g(7) = −1 and f(7) = 3. Find (f/g)(7).

Find the range of y = 4/(x² + 2).

Where are the vertical asymptotes of y = (cos x)/(sin x) (that is, cot x) for 0 ≤ x ≤ 2π?

More practice: warm-up

f(x) = 6x and g(x) = 2. Find (f/g)(x).

f(3) = 12 and g(3) = 4. Find (f/g)(3).

For which x is (f/g)(x) undefined if g(x) = x − 9?

What is the zero of (x − 5)/(x + 1)?

Does y = 1/(x² + 4) have a vertical asymptote?

More practice: on your own

For A(n) = (500 + 8n)/n (the average cost per item of making n items), find A(50), A(500) and the horizontal asymptote. What does it mean?

Find all holes and asymptotes of y = (x² − x − 6)/(x² − 9).

f(x) = 2ˣ and g(x) = x. Describe f/g for x > 0 near 0 and for large x.

State the zeros and vertical asymptotes of y = x/sin x for −2π < x < 2π.

Explain why (sin x)/x is never equal to 1 for x ≠ 0.

Challenge

Build a quotient f/g with a vertical asymptote at x = 1, a hole at x = −2, a zero at x = 3, and horizontal asymptote y = 2.

The function y = 1/(1 + 2⁻ˣ) is a quotient used to model the spread of a new idea through a town. Find its domain, range, y-intercept and both horizontal asymptotes.

For what values of k does y = 1/(x² + k) have exactly two vertical asymptotes? None? Explain.

Check your understanding

Quiz

What is the domain of (f/g)(x) if f(x) = √x and g(x) = x − 4?

f(5) = 0 and g(5) = 2. What can you conclude about f/g at x = 5?

Which function has a hole rather than a vertical asymptote at x = 1?

What is the range of y = 1/(x² + 1)?

For x in radians, what value does (sin x)/x approach as x approaches 0?

Where does y = tan x have vertical asymptotes?

For parents and teachers

Big idea: this lesson completes D2.1 with quotients. The key rule: the domain of f/g leaves out every x where g(x) = 0. Then a zero of g gives either a vertical asymptote or (if f is also 0 there because of a shared factor) a hole.

How to help: ask "What makes the denominator zero?" first, every time. Then ask "Is the numerator also zero there?" to decide asymptote or hole. The cottage-sharing hook is a good conversation starter: ask what happens to the cost per person when more friends come.

Common difficulties:

Cancelling a common factor and forgetting the hole.
Thinking every zero of the denominator gives an asymptote.
Doing the (sin x)/x investigation in degree mode by accident.
Saying the range of a quotient includes its horizontal asymptote value without checking (sometimes the graph crosses a horizontal asymptote, as (sin x)/x crosses y = 0 again and again).

Graphing technology (for example the free Desmos grapher) is a good way to check every sketch here.