OMPT drill — Functions

Function basics

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

Lesson

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 you put in the slot. That last phrase matters more than it looks. $f (5)$ means "put 5 in the slot": $25 - 15 = 10$ . And $f (a + 1)$ means put the whole package $a + 1$ in the slot, brackets and all: everywhere an $x$ appeared.

Two vocabulary words carry the rest of the topic. The domain is the set of inputs the machine accepts; the range is the set of outputs it can produce. On the OMPT, domain questions are the reliable ones, and they almost always come down to the same two hazards: you cannot divide by zero, and you cannot take an even root of a negative number. Find where those hazards live, exclude them, and the domain is whatever survives.

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.