This part reviews the mathematical concepts and techniques for the topics covered in the Math Level 2 Subject Test. A sound understanding of these concepts certainly will improve your score. The techniques discussed may help you save time solving some of the problems without a calculator at all. For problems requiring computational power, techniques are described that will help you use your calculator in the most efficient manner.

Your classroom experience will guide your decisions about how best to use a graphing calculator. If you have been through a secondary mathematics program that attached equal importance to graphical, tabular, and algebraic presentations, then you probably will rely on your graphing calculator as your primary tool to help you find solutions. However, if you went through a more traditional mathematics program, where algebra and algebraic techniques were stressed, it may be more natural for you to use a graphing calculator only after considering other approaches.


• Overview

• Polynomial Functions

• Trigonometric Functions and Their Inverses

• Exponential and Logarithmic Functions

• Rational Functions and Limits

• Miscellaneous Functions

1.1 Overview


A function is a process that changes a set of input numbers into a set of output numbers. Functions are usually specified by equations such as . In this equation x represents an input number while y represents the (unique) corresponding output number. Functions can also have names: in the example, we could name the function f. Then the process could be described as , whereby f takes the input number x, multiplies it by 2, subtracts 1, and takes the square root to produce the output y = f (x).

Taken as a group, the input numbers are called the domain of the function, while the output numbers are called the range. Unless otherwise specified, the domain of a function is all real numbers for which the equation produces outputs that are real numbers. In the example above, the domain is the set x ≥ . since 2x − 1 cannot be negative if y is to be a real number. In this case, the range is the set of all non-negative numbers.

The domain of a function can also be established as part of the definition of a function. For example, even though the domain of the example function f is x ≥ , one could, for example, specify the domain x > 5. Unless a domain is explicitly stated, the domain is assumed to be all real values that produce real numbers as outputs.

A function with a small finite domain can be described by a set of ordered pairs instead of an equation. The first number in the pair is from the domain and the second is the corresponding range value. Consider, for example, the function f consisting of the pairs (0, 2), (1, 1), (2, 3), and (3, 8). In this example, the “process” is not systematic: it simply changes 0 to 2 (f (0) = 2); 1 to 1 (f (1) = 1); 2 to 3 (f (2) = 3); and 3 to 8 (f (3) = 8). The domain of this function consists of 0, 1, 2, and 3, while the range consists of 2, 1, 3, and 8. Functions like this are typically used to illustrate certain properties of functions and are discussed later.

A function is actually a special type of relation. A relation describes the association between two variables. An equation such as x2 + y2 = 4 is one way of defining a relation. All ordered pairs (xy) that satisfy the equations are in the relation. In this case, these pairs form the circle of radius 2 centered at the origin.


Typically, a value of x that must be excluded from the domain of a function makes the denominator zero or makes the value of an expression under a radical less than zero.

Circles are examples of relations that are not functions because some x values (0 in the example) have two y values associated with it) (2 and −2), which violates the uniqueness of the output for a given input. Other than circles, relations that are not functions include ellipses, hyperbolas, and parabolas that open right or left, instead of up or down—in other words, the conic sections discussed in Section 2.3.

Like functions, relations can also be defined using specific ordered pairs. The set R = {(1, 1), (1, 2), (2, 1), (2, 2)} is an example of a relation that is not a function because the x value 1 has two y values associated with it (1 and 2).


1.       If {(3,2),(4,2),(3,1),(7,1),(2,3)} is to be a function, which one of the following must be removed from the set?

           (A)  (3,2)

           (B)  (4,2)

           (C)  (2,3)

           (D)  (7,1)

           (E)  none of the above

2.       For f(x) = 3x2 + 4, g(x) = 2, and h = {(1,1), (2,1), (3,2)},

           (A)  is the only function

           (B)  is the only function

           (C)  and are the only functions

           (D)  and are the only functions

           (E)  f, g, and are all functions

3.       What value(s) must be excluded from the domain of ?

           (A)  –2

           (B)  0

           (C)  2

           (D)  2 and –2

           (E)  no value


Looking for answers? All answers to exercises appear at the end of each section. Resist the urge to peek before trying the problems on your own.