VertexMath · MHF4U · Free worksheet

Rational Functions Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Asymptotes of $\dfrac{ax+b}{cx+d}$ : vertical $x=-\dfrac{d}{c}$ , horizontal $y=\dfrac{a}{c}$ . When the numerator's degree is one higher than the denominator's, divide: the quotient is the oblique asymptote.
Hole or asymptote? Factor first. A factor that cancels gives a hole, and its $y$ -value comes from the simplified function.
Reciprocal $y=\dfrac{1}{f(x)}$ : the zeros of $f$ become vertical asymptotes, and the invariant points are where $f(x)=1$ or $f(x)=-1$ .
Rational equations: multiply every term by the LCD, then reject any root that makes an original denominator zero.
Never multiply an inequality by an expression whose sign you don't know. Get $0$ on one side and use a sign chart; a zero of the denominator always gets a round bracket.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Reciprocals · Asymptotes & Holes · Domain & Sign · Equations · Inequalities

K1 Reciprocal of a Quadratic

The graph of $f(x)=x^2-2x-3$ is shown. Each grid square is 1 unit.

a)
State the equations of the asymptotes of $y=\dfrac{1}{f(x)}$ .
b)
Determine the coordinates of the invariant points of $f$ and $y=\dfrac{1}{f(x)}$ .
c)
Determine the coordinates of the local maximum of $y=\dfrac{1}{f(x)}$ .

Stuck? Peek at a hint

Ask what happens to $\dfrac{1}{f(x)}$ where $f(x)=0$ , where $f(x)=\pm1$ , and at the vertex of $f$ .

K2 Asymptotes & Holes

For each function, state the equations of all asymptotes and the coordinates of any hole.

a)
$f(x)=\dfrac{6x-1}{3x+4}$
b)
$g(x)=\dfrac{x^2-9}{x^2+x-6}$
c)
$h(x)=\dfrac{2x^2+3x-1}{x-1}$

Stuck? Peek at a hint

Factor the numerator and the denominator before looking for any asymptote.

K3 Domain, Range & Sign

a)
State the domain and range of $f(x)=\dfrac{2x-1}{x+3}$ in interval notation.
b)
State the domain and range of $g(x)=\dfrac{x^2-x-2}{x-2}$ in interval notation.
c)
Determine the intervals on which $h(x)=\dfrac{x+2}{(x-1)(x-4)}$ is positive and those on which it is negative.

Stuck? Peek at a hint

Factor and cancel before stating a domain or range. For (c), mark every zero of the numerator and of the denominator on a number line.

K4 Rational Equations

Solve each equation. Check every candidate against the restrictions.

a)
$\dfrac{3}{x-2}=\dfrac{5}{x+1}$
b)
$\dfrac{x}{x-3}-\dfrac{2}{x+1}=\dfrac{12}{(x-3)(x+1)}$
c)
$\dfrac{2}{x}+\dfrac{3}{x+2}=1$

Stuck? Peek at a hint

Multiply every term, including the right side, by the LCD.

K5 Rational Inequalities

Solve each inequality. Write the answer in interval notation.

a)
$\dfrac{x-5}{x+2}\ge 0$
b)
$\dfrac{x+4}{x-1}<3$
c)
$\dfrac{x^2-x-6}{x^2-1}\le 0$

Stuck? Peek at a hint

Get $0$ on one side and combine into a single fraction before testing any interval.

Real World (A)

Work Rates · Average Cost · Medication Levels

A1 Two Pumps

After a spring storm, two pumps drain a flooded underpass in Ottawa. Working together, they drain it in 2 hours. Alone, the older pump takes 3 hours longer than the newer one. Let $t$ be the time, in hours, that the newer pump takes alone.

a)
Write a rational equation that models the situation.
b)
Solve the equation. How long does each pump take alone?
c)
The equation has a second root, $t=-2$ . It satisfies the equation and makes no denominator zero. Should it be rejected? Why?

Stuck? Peek at a hint

In one hour, the newer pump drains $\dfrac{1}{t}$ of the underpass.

A2 Club Hoodies

A school club orders custom hoodies. The supplier charges a $600 setup fee plus $15 per hoodie. For an order of $n$ hoodies, the average cost per hoodie, in dollars, is $A(n)=\dfrac{15n+600}{n}$ .

$n$	$20$	$50$	$100$	$200$
$A(n)$	$45$	$27$	$21$	$18$

a)
State the equation of the horizontal asymptote of $A$ . What does it mean for the club?
b)
How many hoodies must the club order for the average cost to be at most $20?
c)
A member says a big enough order will bring the average cost down to $15. Solve $A(n)=15$ and respond.

Stuck? Peek at a hint

Split the fraction: $\dfrac{15n+600}{n}=15+\dfrac{600}{n}$ .

A3 Medication Window

The concentration of a medication in a patient's blood, in mg/L, $t$ hours after a dose is $C(t)=\dfrac{50t}{t^2+4}$ , for $t\ge0$ . The medication works while the concentration is at least 10 mg/L.

a)
Evaluate $C(1)$ and $C(10)$ , and state the horizontal asymptote of $C$ . What happens to the concentration in the long run?
b)
Solve $C(t)\ge10$ . For how many hours does the medication work?
c)
A doctor wants this dose to reach 15 mg/L. Solve $C(t)=15$ and interpret the result.

Stuck? Peek at a hint

Multiply both sides by $t^2+4$ , which is always positive, so the inequality sign does not change.

Think It Through (T)

Error Analysis · Building Functions

T1 Spot the Flaw

A student analyses $y=\dfrac{1}{f(x)}$ for $f(x)=x^2-2x+5$ :
" $f$ is a quadratic, so $\frac{1}{f}$ has two vertical asymptotes. And $f$ increases on $(1,\infty)$ , so $\frac{1}{f}$ increases there too."

a)
Show that $f$ has no zeros. What does this mean for the vertical asymptotes of $y=\dfrac{1}{f(x)}$ ?
b)
Correct the student's second claim.
c)
State the range of $y=\dfrac{1}{f(x)}$ and the coordinates of its maximum point.

Stuck? Peek at a hint

Complete the square on $f$ .

T2 Build It Backwards

Write the equation of a rational function with each property. In (b) and (c), many answers are possible.

a)
It has the form $y=\dfrac{ax+b}{cx+d}$ and the graph below. The asymptotes are dashed. Each grid square is 1 unit.
b)
A hole at $(1,4)$ and a vertical asymptote $x=-2$ .
c)
A horizontal asymptote $y=1$ , and a graph that crosses this asymptote. Give the crossing point.

Stuck? Peek at a hint

Start from the asymptotes: a vertical asymptote $x=a$ puts a factor $x-a$ in the denominator.

Boss Level

Optional challenge

Exactly One Solution

Consider the equation $\dfrac{x}{x-2}+\dfrac{k}{x+1}=\dfrac{3}{(x-2)(x+1)}$ , where $k$ is a real constant.

a)
Show that every solution satisfies $x^2+(k+1)x-(2k+3)=0$ , and that $x=2$ is never a root of this quadratic.
b)
Find the values of $k$ for which the quadratic has a double root.
c)
Find all values of $k$ for which the original equation has exactly one solution.

Stuck? Peek at a hint

Multiply through by $(x-2)(x+1)$ , and keep track of the restrictions on $x$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) $x=-1$ , $x=3$ and $y=0$
b) $(1-\sqrt{5},1)$ , $(1+\sqrt{5},1)$ , $(1-\sqrt{3},-1)$ , $(1+\sqrt{3},-1)$
c) $\left(1,-\dfrac{1}{4}\right)$

a) VA $x=-\dfrac{4}{3}$ ; HA $y=2$ ; no hole
b) Hole at $\left(-3,\dfrac{6}{5}\right)$ ; VA $x=2$ ; HA $y=1$
c) VA $x=1$ ; oblique asymptote $y=2x+5$ ; no hole

a) $D=(-\infty,-3)\cup(-3,\infty)$ , $R=(-\infty,2)\cup(2,\infty)$
b) $D=(-\infty,2)\cup(2,\infty)$ , $R=(-\infty,3)\cup(3,\infty)$
c) Positive on $(-2,1)\cup(4,\infty)$ ; negative on $(-\infty,-2)\cup(1,4)$

a) $x=\dfrac{13}{2}$
b) $x=-2$ ( $x=3$ is extraneous)
c) $x=-1$ or $x=4$

a) $(-\infty,-2)\cup[5,\infty)$
b) $(-\infty,1)\cup\left(\dfrac{7}{2},\infty\right)$
c) $[-2,-1)\cup(1,3]$

Real World

a) $\dfrac{1}{t}+\dfrac{1}{t+3}=\dfrac{1}{2}$
b) $t=3$ : the newer pump takes 3 h, the older pump 6 h
c) Yes: a pump cannot take $-2$ hours. The root is valid algebra, but outside the domain $t>0$ of the situation.

a) $y=15$ : for very large orders, the average cost gets close to $15 per hoodie
b) $n\ge 120$ : at least 120 hoodies
c) $\dfrac{600}{n}=0$ has no solution: the average cost stays above $15 for every order size.

a) $C(1)=10$ mg/L, $C(10)\approx4.81$ mg/L; $y=0$ : the concentration falls toward 0 as the body clears the drug
b) $1\le t\le4$ , so for 3 hours
c) $3t^2-10t+12=0$ has no real root (discriminant $-44$ ): this dose never reaches 15 mg/L (its peak is 12.5 mg/L, at $t=2$ ).

Think It Through

a) $f(x)=(x-1)^2+4\ge4$ , so $f$ has no zeros and $y=\frac{1}{f(x)}$ has no vertical asymptote (only $y=0$ ).
b) Where $f$ is positive and increasing, $\frac{1}{f}$ decreases: $\frac{1}{f}$ decreases on $(1,\infty)$ and increases on $(-\infty,1)$ .
c) $R=\left(0,\dfrac{1}{4}\right]$ ; maximum at $\left(1,\dfrac{1}{4}\right)$

a) $y=\dfrac{-2x+10}{x-3}$ (or $y=-2+\dfrac{4}{x-3}$ )
b) e.g. $y=\dfrac{12(x-1)}{(x-1)(x+2)}$
c) e.g. $y=\dfrac{x^2+x}{x^2+1}$ , which crosses $y=1$ at $(1,1)$

Boss Level

Boss

a) Multiply by $(x-2)(x+1)$ : $x(x+1)+k(x-2)=3$ . At $x=2$ the quadratic equals $3\ne0$ .
b) $k=-5\pm2\sqrt{3}$
c) $k=-1$ or $k=-5\pm2\sqrt{3}$

Free cheat sheet

Rational Functions Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

Unit Summary

One printable page with every rule.

↓ Unit Summary PDF

Build your own

The same page with blanks: fill it in from memory, then check.

↓ Build-your-own PDF

Phone card

The essentials on one screen. Save it to your photos.

↓ Phone card (image)

Most testedFind the asymptotes, hole and intercepts of $f(x)=\dfrac{x^2-x-2}{x^2-4}$ . (card 3)

1Reciprocal functions $1/f(x)$

UseZeros of $f$ become vertical asymptotes. Invariant points: $f(x)=\pm1$ . Where $f$ increases, $1/f$ decreases. A vertex $(h,k)$ of $f$ becomes a turning point $(h,\ 1/k)$ .

Example $f(x)=x^2-4$ : $1/f$ has asymptotes $x=\pm2$ and $y=0$ , and a turning point at $(0,\ -1/4)$ .

TrapFor $y=1/f(x)$ , invariant points solve $f(x)=\pm1$ ; $f(x)=0$ gives the asymptotes.

Trap $1/(x^2+1)$ has no vertical asymptote: $x^2+1$ is never $0$ .

2Asymptotes

$y=\dfrac{ax+b}{cx+d}$ : VA $x=-\dfrac dc$ , HA $y=\dfrac ac$

UseVertical asymptotes: zeros of the denominator that do not cancel. Horizontal: lower numerator degree, $y=0$ ; equal, the ratio of leading coefficients; higher, none.

TrapEqual degrees give HA $y=a/c$ : $(2x+1)/(x-3)$ has $y=2$ , not $y=1$ .

3Holes and oblique asymptotes

UseFactor first. A factor that cancels gives a hole; take its $y$ from the simplified form. Numerator degree one higher: divide, and the quotient is the oblique asymptote.

Example $\dfrac{x^2-x-2}{x^2-4}=\dfrac{x+1}{x+2}$ , $x\ne2$ : hole $(2,\ 3/4)$ , VA $x=-2$ , HA $y=1$ .

4Domain, range and sign

UseDomain: remove asymptotes and holes. Range: also remove a hole's $y$ -value (if not reached elsewhere). Sign chart: zeros of the numerator and the denominator.

TrapA graph may cross its horizontal asymptote; it never crosses a vertical one.

5Rational equations

UseMultiply every term by the LCD. Reject any root that makes an original denominator $0$ . Work problems: add rates, $1/a+1/b=1/t$ .

Example $x/(x-2)+3=2/(x-2)$ : $x+3(x-2)=2$ , so $x=2$ : rejected, no solution.

TrapTwo pipes taking $t$ and $t+3$ hours: add the rates $1/t+1/(t+3)$ , never the times.

6Rational inequalities

UseGet $0$ on one side as one fraction. Critical values: zeros of the numerator and of the denominator. Test each interval. Denominator zeros always get round brackets.

Example $(x-3)/(x+1)\le0$ : $(-1,\ 3]$ .

TrapNever multiply both sides by an expression whose sign you don't know: use a sign chart.

Hole or asymptote?

A factor cancels Hole

A factor stays in the denominator Vertical asymptote

Numerator degree one higher Oblique asymptote

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Core Skills (K)

K1 Reciprocal of a Quadratic

K2 Asymptotes & Holes

K3 Domain, Range & Sign

K4 Rational Equations

K5 Rational Inequalities

Real World (A)

A1 Two Pumps

A2 Club Hoodies

A3 Medication Window

Think It Through (T)

T1 Spot the Flaw

T2 Build It Backwards

Boss Level

Exactly One Solution

Answer key

Core Skills

Real World

Think It Through

Boss Level

Rational Functions Cheat Sheet

Unit Summary

Build your own

Phone card

1Reciprocal functions 1/f(x)1/f(x)1/f(x)

2Asymptotes

3Holes and oblique asymptotes

4Domain, range and sign

5Rational equations

6Rational inequalities

Hole or asymptote?

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What is VertexMath?

1Reciprocal functions $1/f(x)$