OMPT drill — Functions

OMPT function basics practice: lesson and 10 problems

Q: Are functions on the OMPT-A?

Yes. Functions is one of the eight OMPT-A chapters, so roughly 4 to 6 of the 52 questions come from it and more use it as a step. Expect to evaluate f at a number or an expression, solve f(x) = c backwards, state a domain from a root or a denominator, and check whether a point lies on a graph. The later chapters on exponential functions and differentiation build on the same notation.

Q: How do I find the domain of a function for the OMPT?

Look for the two hazards. Every denominator must be non-zero, so solve it equal to zero and exclude every solution. Everything under an even root must be at least zero, so solve that inequality. When both appear, intersect the conditions. A root inside a denominator gives a strict inequality: the domain of 1 over the square root of x minus 2 is x greater than 2, not greater than or equal.

Q: What is the difference between f(a + 1) and f(a) + 1 on the OMPT?

f(a + 1) changes the input: the whole package a + 1 goes into every x slot, with brackets, and you expand afterwards. f(a) + 1 leaves the input alone and adds 1 to the output. With f(x) = x squared, f(3) + 1 is 10 while f(4) is 16. The OMPT asks for the first kind far more often, and dropped brackets are the usual reason it goes wrong.

Q: Is function basics on the OMPT-C?

No. The OMPT-C syllabus lists numbers, algebra, linear formulas and equations, and geometry, with no Functions chapter. Lines and linear formulas do appear, so the notation y = ax + b matters, but domain, range and f(x) substitution are not examined. Put the time into linear equations and geometry.

Q: How do I solve f(x) = c on the OMPT when f is quadratic?

Treat it as an equation, not a substitution. Move everything to one side so it reads quadratic equals zero, factorise or use the formula, and give every solution. f(x) = x squared minus 3x with f(x) = 10 becomes x squared minus 3x minus 10 = 0, which factorises to (x - 5)(x + 2), so x = 5 or x = -2. A "how many solutions" version is answered by comparing c with the vertex height.

Q: Do I need a calculator for function questions on the OMPT?

No. The maths variants do not provide one unless a question supplies it, and the numbers are chosen so that substitution stays in whole numbers and small fractions. If you are evaluating a function and the arithmetic explodes, you have most likely dropped a bracket or substituted into the wrong place. Type exact values: a fraction stays a fraction.

Domain, range, and reading f(x) notation without flinching. Domain restrictions from roots and denominators are a reliable source of test questions. Attempt each problem before opening the solution; the notation only becomes automatic by writing it yourself.

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

OMPT function basics: the lesson

What the OMPT tests in function basics

A function is a machine with one strict promise: feed it an input and it returns exactly one output. The notation $f (x) = x^{2} - 3 x$ names the machine $f$ and describes what it does to whatever sits in the slot. $f (5)$ means "put 5 in the slot": $25 - 15 = 10$ . $f (a + 1)$ means put the whole package $a + 1$ in the slot, brackets and all, everywhere an $x$ appeared. The Functions chapter sits on OMPT-A, B, D, E, F and G, and it feeds every later function topic, so the exam returns to these moves again and again.

Four sub-skills cover nearly every question. Evaluating: compute $f (- 2)$ or simplify $f (a + 1)$ . Solving backwards: find every $x$ with $f (x) = 10$ . Domain: say which inputs are legal, which on the OMPT comes down to two hazards, division by zero and an even root of a negative number. Graph reading: decide whether a point lies on the curve, or how many times a horizontal line meets it. Domain and substitution are the heavy pair. In my sessions the marks go missing on dropped brackets and on $>$ against $\geq$ , not on anything deep.

Notation and answer format the OMPT expects

Values are typed as exact numbers: $f (- 2) = 10$ , or a fraction such as $3/4$ , never a rounded decimal unless the question says "round to".
A domain is typed as an inequality or a set of exclusions: $x \geq 3$ , or "all real numbers except $x = 3$ and $x = - 3$ ". Roots give $\geq$ , denominators give $\neq =$ , a root in a denominator gives a strict $>$ .
If $f (x) = c$ has two solutions, give both. If it has none, the answer is the phrase "no solution", not 0.
A simplified expression such as $a^{2} - a - 2$ is typed with the powers written out; the platform checks it algebraically, so a different order of terms is accepted.
A point on a graph is a pair $(2, 3)$ , with the input first.

Worked example 1: substituting an expression

Given $f (x) = x^{2} - 3 x$ , find and simplify $f (a + 1)$ .
Replace every $x$ with $(a + 1)$ , keeping the brackets: $f (a + 1) = (a + 1)^{2} - 3 (a + 1)$ .
Expand each piece: $(a + 1)^{2} = a^{2} + 2 a + 1$ and $3 (a + 1) = 3 a + 3$ .
Combine: $a^{2} + 2 a + 1 - 3 a - 3 = a^{2} - a - 2$ .
Check with $a = 1$ : the direct route gives $f (2) = 4 - 6 = - 2$ , the formula gives $1 - 1 - 2 = - 2$ . They agree. The trap is writing $a + 1^{2}$ or applying the $- 3$ to $a$ only.

Worked example 2: a root in the formula

Find the domain of $f (x) = 2 x - 6$ .
The hazard is the square root: whatever sits under it must be at least zero. Require $2 x - 6 \geq 0$ .
Solve: $2 x \geq 6$ , so $x \geq 3$ . That inequality is the domain.
Test it: $f (3) = 0 = 0$ is legal, $f (5) = 4 = 2$ is fine, and $f (1) = - 4$ is not a real number.
The boundary is included. Write $x \geq 3$ , not $x > 3$ . Strict against inclusive is exactly what the examiner is checking.

The full lesson works through solving $f (x) = c$ backwards, denominators with two zeros, a root and a fraction in one formula, range questions, a parameter fixed by one function value, and the "how many solutions" question read off a graph. It closes with the edge cases the OMPT likes and the exact slips I mark most often.

How each OMPT variant tests function basics

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 4 to 6 of 52 questions
Official chapter: Quadratic equations and functions, 22% of the paper.
Short numeric answers: evaluate f at a number or an expression, solve f(x) = c backwards, and state a domain from a root or a denominator. One graph-reading item is usual.
OMPT-B
64 questions
Testedroughly 4 to 6 of 64 questions
Official chapter: Quadratic equations and functions, 10% of the paper.
Same shapes as OMPT-A, plus domain questions with two hazards at once and function values that feed later differentiation and integration items.
OMPT-C
30 questions
Not tested
OMPT-C has no Functions chapter; only linear formulas appear, through the Linear formulas and equations chapter. Spend the time on linear equations, lines 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.
Domain and substitution appear as the opening parts of a long question, often with a parameter to fix from one function value before the calculus starts.
OMPT-E
40 questions
Testedroughly 2 to 4 of 40 questions
Official chapter: Quadratic equations and functions, 10% of the paper.
Evaluate, solve f(x) = c and read a graph. Domain questions are lighter here; the function ideas mostly support the statistics chapters.
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.
Multi-part business questions: evaluate a cost or revenue function, state where it is defined, then differentiate it. Substituting an expression like f(q + 1) turns up.
OMPT-G
26 questions
Testedroughly 2 to 3 of 26 questions
Official chapter: Functions, graphs and equations, 45% of the paper.
Function notation and domains applied to physics and chemistry formulas: evaluate at a time, solve for the input, and read values off a given graph.

OMPT function basics 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

Let $g (x) = x^{2} - 3 x$ . Compute $g (- 2)$ .

Show the worked solution

Answer: 10

I substitute carefully with brackets: $g (- 2) = (- 2)^{2} - 3 (- 2) = 4 + 6 = 10$ . The brackets matter twice here — squaring the negative gives $+ 4$ , and subtracting a negative product adds 6.

Problem 2Multiple choice

What is the domain of $f (x) = x - 4$ ?

A $x \geq 4$
B $x > 4$
C $x \geq 0$
Dall real numbers

Show the worked solution

Answer: A — $x \geq 4$

A square root needs a non-negative input, so I require $x - 4 \geq 0$ , which gives $x \geq 4$ . The boundary $x = 4$ is included, since $0 = 0$ is perfectly fine.

Problem 3Numeric answer

Let $f (x) = 2 x + 5$ . Solve $f (t) = 17$ for $t$ .

Show the worked solution

Answer: 6

Solving $f (t) = 17$ means finding the input that produces output 17: $2 t + 5 = 17$ , so $2 t = 12$ and $t = 6$ . This is the reverse question of evaluation — I am asked for the input, not the output.

Problem 4Multiple choice

Which point lies on the graph of $h (x) = x^{2} - 1$ ?

A $(- 3, 8)$
B $(2, 5)$
C $(1, 1)$
D $(3, 10)$

Show the worked solution

Answer: A — $(- 3, 8)$

A point $(a, b)$ lies on the graph exactly when $h (a) = b$ . Testing: $h (- 3) = 9 - 1 = 8$ , so $(- 3, 8)$ is on the graph. The others fail: $h (2) = 3$ , $h (1) = 0$ , $h (3) = 8$ .

Problem 5Spot the error

A student finds the domain of $f (x) = \frac{1}{x ^{2} - 9}$ . Step 1: the denominator may not be zero. Step 2: $x^{2} - 9 = 0$ when $x = 9$ . Step 3: domain is all reals except $x = 9$ . Which step contains the error?

Show the worked solution

Answer: Step 2

Step 1 states the right principle. Step 2 solves the equation wrong: $x^{2} - 9 = 0$ means $x^{2} = 9$ , so $x = 3$ or $x = - 3$ , not $x = 9$ . The domain is all real numbers except $\pm 3$ . At $x = 9$ the function is fine: $1/ (81 - 9) = 1/72$ .

Problem 6Numeric answer

What is the minimum value of $f (x) = x^{2} + 2$ ?

Show the worked solution

Answer: 2

The square $x^{2}$ is never negative and equals zero at $x = 0$ . So the smallest $f$ can be is $0 + 2 = 2$ , reached at $x = 0$ . Every other input adds a positive square on top of 2.

Problem 7Numeric answer

Let $f (x) = 3 x^{2} - 2 x + 1$ . Compute $f (- 1) - f (2)$ .

Show the worked solution

Answer: -3

Two substitutions, each with brackets. $f (- 1) = 3 (- 1)^{2} - 2 (- 1) + 1 = 3 + 2 + 1 = 6$ . $f (2) = 3 \cdot 4 - 4 + 1 = 12 - 4 + 1 = 9$ . The difference is $6 - 9 = - 3$ . Check the first value once more: the square of $- 1$ is $+ 1$ , and subtracting $- 2$ adds 2, so 6 stands.

Problem 8Multiple choice

What is the domain of $h (x) = \frac{5 - x}{x + 2}$ ?

A $x \leq 5$
B $x \leq 5$ and $x \neq = - 2$
C $x < 5$ and $x \neq = - 2$
D $x \geq 5$ and $x \neq = - 2$

Show the worked solution

Answer: B — $x \leq 5$ and $x \neq = - 2$

Two hazards. The root needs $5 - x \geq 0$ , so $x \leq 5$ , with 5 itself allowed since $0 = 0$ . The denominator needs $x + 2 \neq = 0$ , so $x \neq = - 2$ . Both conditions together: $x \leq 5$ and $x \neq = - 2$ . Check the edges: $h (5) = 0/7 = 0$ is legal, $h (- 2)$ divides by zero, and $h (6)$ needs $- 1$ .

Problem 9Numeric answer

The function $f (x) = x^{2} + k x - 6$ satisfies $f (3) = 0$ . For which other value of $x$ is $f (x) = 0$ ?

Show the worked solution

Answer: -2

First pin down $k$ : $f (3) = 9 + 3 k - 6 = 3 k + 3 = 0$ , so $k = - 1$ . Now $f (x) = x^{2} - x - 6 = (x - 3) (x + 2)$ , and the zeros are $x = 3$ and $x = - 2$ . The other one is $- 2$ . Check: $f (- 2) = 4 + 2 - 6 = 0$ . A shortcut: the product of the zeros of $x^{2} + k x - 6$ is $- 6$ , so the partner of 3 is $- 2$ whatever $k$ turns out to be.

Problem 10Multiple choice

Let $f (x) = x + 3 - x - 1$ . How many real solutions does the equation $f (x) = 0$ have?

Show the worked solution

Answer: B — 1

Setting $f (x) = 0$ gives $x + 3 = x + 1$ . Squaring: $x + 3 = x^{2} + 2 x + 1$ , so $x^{2} + x - 2 = 0$ , which factorises as $(x + 2) (x - 1) = 0$ , candidates $x = - 2$ and $x = 1$ . Squaring can invent solutions, so I test both in the original. $x = 1$ : $4 = 2$ and $1 + 1 = 2$ , correct. $x = - 2$ : $1 = 1$ but $- 2 + 1 = - 1$ , and a square root is never negative, so this one is rejected. Exactly one solution.

OMPT function basics: questions students ask

Are functions on the OMPT-A?

Yes. Functions is one of the eight OMPT-A chapters, so roughly 4 to 6 of the 52 questions come from it and more use it as a step. Expect to evaluate f at a number or an expression, solve f(x) = c backwards, state a domain from a root or a denominator, and check whether a point lies on a graph. The later chapters on exponential functions and differentiation build on the same notation.

How do I find the domain of a function for the OMPT?

Look for the two hazards. Every denominator must be non-zero, so solve it equal to zero and exclude every solution. Everything under an even root must be at least zero, so solve that inequality. When both appear, intersect the conditions. A root inside a denominator gives a strict inequality: the domain of 1 over the square root of x minus 2 is x greater than 2, not greater than or equal.

What is the difference between f(a + 1) and f(a) + 1 on the OMPT?

f(a + 1) changes the input: the whole package a + 1 goes into every x slot, with brackets, and you expand afterwards. f(a) + 1 leaves the input alone and adds 1 to the output. With f(x) = x squared, f(3) + 1 is 10 while f(4) is 16. The OMPT asks for the first kind far more often, and dropped brackets are the usual reason it goes wrong.

Is function basics on the OMPT-C?

No. The OMPT-C syllabus lists numbers, algebra, linear formulas and equations, and geometry, with no Functions chapter. Lines and linear formulas do appear, so the notation y = ax + b matters, but domain, range and f(x) substitution are not examined. Put the time into linear equations and geometry.

How do I solve f(x) = c on the OMPT when f is quadratic?

Treat it as an equation, not a substitution. Move everything to one side so it reads quadratic equals zero, factorise or use the formula, and give every solution. f(x) = x squared minus 3x with f(x) = 10 becomes x squared minus 3x minus 10 = 0, which factorises to (x - 5)(x + 2), so x = 5 or x = -2. A "how many solutions" version is answered by comparing c with the vertex height.

Do I need a calculator for function questions on the OMPT?

No. The maths variants do not provide one unless a question supplies it, and the numbers are chosen so that substitution stays in whole numbers and small fractions. If you are evaluating a function and the arithmetic explodes, you have most likely dropped a bracket or substituted into the wrong place. Type exact values: a fraction stays a fraction.

Test hubs that cover function basics:OMPT-A OMPT-B OMPT-D OMPT-E OMPT-F OMPT-G All 28 topics

OMPT function basics practice: lesson and 10 problems

OMPT function basics: the lesson

What the OMPT tests in function basics

Notation and answer format the OMPT expects

Worked example 1: substituting an expression

Worked example 2: a root in the formula

How each OMPT variant tests function basics

OMPT function basics practice problems

OMPT function basics: questions students ask

Related OMPT topics and test hubs