OMPT drill — Functions

OMPT rational functions practice: lesson and 10 problems

Fractional functions: asymptotes, excluded values, and solving equations with x in the denominator, where multiplying through without checking creates phantom solutions. Attempt each problem before opening the solution; the notation only becomes automatic by writing it yourself.

Tested inOMPT-A OMPT-B OMPT-D OMPT-F

OMPT rational functions: the lesson

What the OMPT tests in rational functions

A rational function is a fraction of polynomials, such as $f (x) = \frac{2 x + 1}{x - 3}$ , and it belongs to the Functions chapter of OMPT-A, B, D and F. The sub-skills are: stating the domain (which inputs make the denominator zero), reading the vertical and horizontal asymptotes off the formula, simplifying a rational expression by cancelling a common factor, adding two fractions with $x$ downstairs, and solving an equation with $x$ in a denominator. That last family is the heavy one. A typical question reads "Solve $\frac{x + 2}{x - 1} = 3$ " or "For which $q$ is the average cost 4 euros?", and a fair share of them contain a candidate that the denominator refuses.

On OMPT-A and OMPT-B I expect one or two rational-function items among the four to six Functions questions. On OMPT-D and OMPT-F the fraction appears inside a longer question, often as an average-cost model or as the function whose derivative you are about to take, so the algebra here decides marks in other chapters as well.

Notation and answer format the OMPT expects

A vertical asymptote is a line, so type it as x = 3, and a horizontal asymptote as y = 2. A domain is typed as a restriction: all real x except 3, or $x \neq = 3$ .
Exact answers: 5/2, not 2.5, unless the question says "round to". A solution containing a root is typed as sqrt(2).
When both candidates survive, give both. When the only candidate is a banned value, the answer is "no solution", and the box accepts that phrase.
A simplified expression is typed as a single fraction with factors cancelled, such as (x - 2)/x, with the condition $x \neq = - 2$ stated when the question asks for it.

Worked example 1: full asymptote analysis

Find the domain and the asymptotes of $f (x) = \frac{2 x + 1}{x - 3}$ .
Domain: the denominator is zero at $x = 3$ , so the domain is all real $x$ except 3.
Vertical asymptote: $x = 3$ , provided the numerator is not also zero there. $2 (3) + 1 = 7 \neq = 0$ , so it is a genuine asymptote and not a hole.
Horizontal asymptote: for enormous $x$ the $+ 1$ and the $- 3$ stop mattering and the fraction behaves like $\frac{2 x}{x} = 2$ . So $y = 2$ .
Feel-check with a big input: $f (1000) = \frac{2001}{997} \approx 2.007$ , just above 2 as predicted. The trap is reading the horizontal asymptote as the ratio of constants, $\frac{1}{- 3}$ ; it is the ratio of leading coefficients.

Worked example 2: solving with the ban list

Solve $\frac{x + 2}{x - 1} = 3$ .
Ban list first: $x \neq = 1$ . I write it before any algebra, every time.
Multiply both sides by $(x - 1)$ : $x + 2 = 3 (x - 1) = 3 x - 3$ .
Collect: $5 = 2 x$ , so $x = \frac{5}{2}$ .
Compare with the ban list: $\frac{5}{2} \neq = 1$ , so it survives. Verify: $\frac{5/2 + 2}{5/2 - 1} = \frac{9/2}{3/2} = 3$ . The mark is lost by students who multiply only the $x$ by 3 and not the $- 1$ .

The full lesson covers holes versus asymptotes, the three horizontal-asymptote cases by degree, intercepts and sketching, the phantom solution, an equation with two fractions to combine, a rational inequality, a speed word problem and a parameter question about when an equation has no solution. It ends with the edge cases the OMPT likes and the slips I mark most often.

How each OMPT variant tests rational functions

The chapter weights are the official ones from the omptest.org learning-assessment pages. The question counts are my estimates of how much of that chapter this one topic takes up. The 21-question papers (OMPT-D and OMPT-F) ask long multi-part questions, so a topic there usually shows up as a step inside a bigger problem.

OMPT-A
52 questions
Testedroughly 1 to 2 of 52 questions, inside the 4 to 6 for the Functions chapter
Official chapter: Quadratic equations and functions, 22% of the paper.
State the domain or an asymptote from the formula, simplify a fraction by cancelling a factor, and solve one equation with x in the denominator, usually with a phantom candidate planted. Exact answers such as 5/2 or "no solution".
OMPT-B
64 questions
Testedroughly 1 to 2 of 64 questions
Official chapter: Quadratic equations and functions, 10% of the paper.
Same shapes as OMPT-A, and the rational function often returns later as the thing to differentiate with the quotient rule or to integrate after splitting.
OMPT-C
30 questions
Not tested
OMPT-C has no Functions chapter; fractions with x downstairs do not appear beyond simple algebra. Spend the time on linear equations and geometry instead.
OMPT-D
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, equations and inequalities, 30% of the paper.
Asymptotes and domain feed a limits part, then a derivative part, in one long question. Expect a hole versus asymptote decision and a rational inequality solved with a sign chart.
OMPT-E
40 questions
Not tested
OMPT-E has a Functions chapter but its emphasis is linear, quadratic and exponential models for the statistics that follow; fractional functions are not a named skill there. Study function basics and exponentials instead.
OMPT-F
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, 15% of the paper.
Business framing: average cost C(q) = (fixed + variable)/q, its horizontal asymptote as the long-run cost per unit, and solving for the output that hits a target cost. A parameter version is common.
OMPT-G
26 questions
Not tested
OMPT-G has a Functions chapter aimed at linear, quadratic and exponential models for physics and chemistry; rational functions are not a named skill. Put the time into trigonometry and geometry.

OMPT rational functions practice problems

10 original problems in the formats the OMPT uses: typed numeric answers, multiple choice, and spot-the-error. The last few are harder than the exam average on purpose. Solutions and the common mistake sit behind each problem.

Problem 1Numeric answer

The function $f (x) = \frac{3}{x - 5}$ has a vertical asymptote at $x = a$ . What is $a$ ?

Show the worked solution

Answer: 5

A vertical asymptote appears where the denominator hits zero while the numerator does not: $x - 5 = 0$ gives $x = 5$ , and the numerator there is 3. So $a = 5$ . Near that value the outputs blow up — $f (5.01) = 300$ , for instance.

Problem 2Multiple choice

What is the horizontal asymptote of $g (x) = \frac{2 x + 1}{x - 4}$ ?

A $y = 2$
B $y = - 4$
C $y = 0$
D $y = \frac{1}{4}$

Show the worked solution

Answer: A — $y = 2$

Numerator and denominator have the same degree, so for very large $x$ the fraction behaves like the ratio of leading coefficients: $2 x / x = 2$ . The asymptote is $y = 2$ . Trying $x = 1000$ : $2001/996 \approx 2.01$ , already hugging it.

Problem 3Numeric answer

Solve for $w$ : $\frac{12}{w + 2} = 3$ .

Show the worked solution

Answer: 2

I multiply both sides by $w + 2$ : $12 = 3 (w + 2)$ , so $4 = w + 2$ and $w = 2$ . The value does not make the denominator zero ( $2 + 2 = 4$ ), so it is a genuine solution: $12/4 = 3$ .

Problem 4Multiple choice

Simplify $\frac{x ^{2} - 4}{x ^{2} + 2 x}$ as far as possible.

A $\frac{x - 2}{x}$
B $\frac{x + 2}{x}$
C $\frac{- 4}{2 x}$
D $x - 2$

Show the worked solution

Answer: A — $\frac{x - 2}{x}$

I factor top and bottom: $\frac{( x - 2 ) ( x + 2 )}{x ( x + 2 )}$ . The common factor $(x + 2)$ cancels (valid for $x \neq = - 2$ ), leaving $\frac{x - 2}{x}$ . A numeric check at $x = 4$ : original $12/24 = 0.5$ , simplified $2/4 = 0.5$ .

Problem 5Spot the error

A student simplifies $\frac{n ^{2} + 5}{n ^{2}}$ . Step 1: cancel $n^{2}$ above and below. Step 2: result is 5. Step 3: check at $n = 1$ : original gives 6, simplified gives 5 — "close enough". Which step contains the error?

Show the worked solution

Answer: Step 1

Step 1 cancels $n^{2}$ even though it is a term of the numerator, not a factor of it. The honest simplification splits the fraction: $\frac{n ^{2} + 5}{n ^{2}} = 1 + \frac{5}{n ^{2}}$ . Step 3 actually caught the crime — at $n = 1$ the original is 6 and the "simplified" version is 5 — but the student waved it away. When a check fails, the algebra is wrong; there is no "close enough".

Problem 6Numeric answer

A workshop's average cost per unit is $C (q) = \frac{50 + 2 q}{q}$ euros when it produces $q$ units. For which $q$ is the average cost exactly 4 euros?

Show the worked solution

Answer: 25

I set $\frac{50 + 2 q}{q} = 4$ and multiply by $q$ : $50 + 2 q = 4 q$ , so $50 = 2 q$ and $q = 25$ . Checking: $(50 + 50) /25 = 4$ euros per unit, exactly as required.

Problem 7Multiple choice

What is the domain of $f (x) = \frac{x + 1}{x ^{2} - 5 x + 6}$ ?

AAll real $x$ except 2 and 3
BAll real $x$ except $- 1$
CAll real $x$ except $- 2$ and $- 3$
DAll real $x$ except $- 1$ , 2 and 3

Show the worked solution

Answer: A — All real $x$ except 2 and 3

The domain excludes only the inputs that make the denominator zero. I factorise: $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ , zero at $x = 2$ and $x = 3$ . The numerator plays no part in the domain; $x = - 1$ is a perfectly good input, it just gives the output 0. Check: $f (- 1) = 0/12 = 0$ , while $f (2)$ would divide by $4 - 10 + 6 = 0$ .

Problem 8Numeric answer

Solve for $x$ : $\frac{4}{x - 1} = \frac{6}{x + 2}$ .

Show the worked solution

Answer: 7

Ban list: $x \neq = 1$ and $x \neq = - 2$ . Both sides are single fractions, so I cross-multiply: $4 (x + 2) = 6 (x - 1)$ , giving $4 x + 8 = 6 x - 6$ , so $14 = 2 x$ and $x = 7$ . It is on neither ban. Check: $4/6 = 2/3$ and $6/9 = 2/3$ .

Problem 9Numeric answer

The equation $\frac{1}{x} + \frac{1}{x + 3} = \frac{1}{2}$ has two solutions. Give their sum.

Show the worked solution

Answer: 1

Ban list: $x \neq = 0$ , $x \neq = - 3$ . I combine the left side over the common denominator: $\frac{( x + 3 ) + x}{x ( x + 3 )} = \frac{2 x + 3}{x ( x + 3 )}$ . Cross-multiplying with $\frac{1}{2}$ : $2 (2 x + 3) = x (x + 3)$ , so $4 x + 6 = x^{2} + 3 x$ and $x^{2} - x - 6 = 0$ , which factorises as $(x - 3) (x + 2) = 0$ . Candidates 3 and $- 2$ , neither banned. Check $x = 3$ : $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$ . Check $x = - 2$ : $- \frac{1}{2} + 1 = \frac{1}{2}$ . The sum is $3 + (- 2) = 1$ .

Problem 10Numeric answer

For which value of $k$ does the equation $\frac{3 x + k}{x - 4} = 2$ have no solution?

Show the worked solution

Answer: -12

Ban list: $x \neq = 4$ . Clearing the denominator gives $3 x + k = 2 x - 8$ , so $x = - 8 - k$ . For every $k$ the algebra produces exactly one candidate, and the equation has no solution only when that candidate is the banned value: $- 8 - k = 4$ , so $k = - 12$ . Check: with $k = - 12$ the left side is $\frac{3 x - 12}{x - 4} = \frac{3 ( x - 4 )}{x - 4} = 3$ for every allowed $x$ , and 3 is never 2.

OMPT rational functions: questions students ask

Are rational functions on the OMPT-A?

Yes, inside the Functions chapter. OMPT-A asks for the domain of a fraction of polynomials, its vertical and horizontal asymptotes, simplifying by cancelling a common factor, and solving an equation with x in the denominator. Expect one or two such items among the four to six Functions questions. OMPT-B, D and F test the same skills; OMPT-C, E and G do not name them.

How do I find the asymptotes of a rational function for the OMPT?

Vertical asymptotes sit where the denominator is zero and the numerator is not; if both are zero the graph has a hole instead. For the horizontal asymptote compare degrees: numerator degree smaller gives y = 0, equal degrees give the ratio of leading coefficients, and a larger numerator degree gives no horizontal asymptote. For (2x + 1)/(x - 3) that means x = 3 and y = 2.

What is a phantom solution and how do I avoid it on the OMPT?

A phantom is a value the algebra produces after you multiply through by a denominator, but which makes that denominator zero in the original equation. It looks like a solution and is not one. Write the banned values before any algebra, and compare every candidate with that list at the end. If the only candidate is banned, the answer is "no solution", which the OMPT accepts.

Can a graph cross its asymptote on the OMPT?

It can cross a horizontal asymptote, because that line only describes what happens far to the left and right. It can never touch a vertical asymptote, because that input is excluded from the domain. To check whether a graph crosses its horizontal asymptote, set the function equal to the asymptote value and solve; no solution means no crossing.

How do I solve a rational inequality on the OMPT?

Do not multiply by the denominator, because its sign is unknown and the direction of the inequality would be unknown too. Instead find where the numerator is zero and where the denominator is zero, use those points to cut the number line into regions, test one value in each region, and read off the regions with the right sign. Include a numerator zero when the inequality is non-strict; always exclude a denominator zero.

Test hubs that cover rational functions:OMPT-A OMPT-B OMPT-D OMPT-F All 28 topics

OMPT rational functions practice: lesson and 10 problems

OMPT rational functions: the lesson

What the OMPT tests in rational functions

Notation and answer format the OMPT expects

Worked example 1: full asymptote analysis

Worked example 2: solving with the ban list

How each OMPT variant tests rational functions

OMPT rational functions practice problems

OMPT rational functions: questions students ask

Related OMPT topics and test hubs