## Calculus AB and Calculus BC

## CHAPTER 1 Functions

**Concepts and Skills**

In this chapter you will review precalculus topics. Although these topics are not directly tested on the AP exam, reviewing them will reinforce some basic principles:

• general properties of functions: domain, range, composition, inverse;

• special functions: absolute value, greatest integer; polynomial, rational, trigonometric, exponential, and logarithmic;

and the BC topics,

• parametrically defined curves • polar curves

**Function**

**Domain**

**Range**

### A. DEFINITIONS

**A1.** A *function f* is a correspondence that associates with each element *a* of a set called the *domain* one and only one element *b* of a set called the *range.* We write

*f* (*a*) = *b*

to indicate that *b* is the *value* of *f* at *a.* The elements in the domain are called *inputs*, and those in the range are called *outputs.*

A function is often represented by an equation, a graph, or a table.

A vertical line cuts the graph of a function in at most one point.

**EXAMPLE 1**

The domain of *f* (*x*) = *x*^{2} − 2 is the set of all real numbers; its range is the set of all reals greater than or equal to −2. Note that

**EXAMPLE 2**

Find the domains of:

**SOLUTIONS:**

**(a)** The domain of is the set of all reals except *x* = 1 (which we shorten to “*x* ≠ 1”).

**(b)** The domain of

**(c)** The domain of is *x* 4, *x* ≠ 0 (which is a short way of writing {*x* | *x* is real, *x* < 0 or 0 < *x* 4}).

**A2.** Two functions *f* and *g* with the same domain may be combined to yield their sum and difference: *f* (*x*) + *g*(*x*) and *f* (*x*) − *g*(*x*), also written as (*f* + *g*) (*x*) and (*f* − *g*) (*x*), respectively; or their product and quotient: *f* (*x*)*g*(*x*) and *f* (*x*)/*g*(*x*), also written as (*fg*)(*x*) and (*f*/*g*) (*x*), respectively. The quotient is defined for all *x* in the shared domain except those values for which *g*(*x*), the denominator, equals zero.

**EXAMPLE 3**

If *f* (*x*) = *x*^{2} − 4*x* and *g*(*x*) = *x* + 1, then find

**SOLUTIONS:**

**Composition**

**A3.** The *composition* (or *composite*) of *f* with *g*, written as *f* (*g*(*x*)) and read as “*f* of *g* of *x*,” is the function obtained by replacing *x* wherever it occurs in *f* (*x*) by *g*(*x*). We also write (*f* ο *g*) (*x*) for *f* (*g*(*x*)). The domain of (*f* ο *g*) (*x*) is the set of all *x* in the domain of *g* for which *g*(*x*) is in the domain of *f*.

**EXAMPLE 4A**

If *f* (*x*) = 2*x* − 1 and *g*(*x*) = *x*^{2}, then does *f* (*g*(*x*)) = *g*(*f* (*x*))?

In general, *f* (*g*(*x*)) ≠ *g*(*f* (*x*)).

**EXAMPLE 4B**

If *f* (*x*) = 4*x*^{2} − 1 and *g*(*x*) = find *f* (*g*(*x*)) and *g*(*f* (*x*)).

**SOLUTIONS:** *f* (*g*(*x*)) = 4*x* − 1 (*x* ≥ 0);

**Symmetry**

**A4.** A function *f* is if, for all *x* in the domain of *f*,

The graph of an odd function is symmetric about the origin; the graph of an even function is symmetric about the *y*-axis.

**EXAMPLE 5**

The graphs of *x*^{3} and *g*(*x*) = 3*x*^{2} − 1 are shown in Figure N1–1; *f* (*x*) is odd, *g*(*x*) even.

**FIGURE N1–1**

**A5.** If a function *f* yields a single output for each input and also yields a single input for every output, then *f* is said to be *one-to-one.* Geometrically, this means that any horizontal line cuts the graph of *f* in at most one point. The function sketched at the left in Figure N1–1 is one-to-one; the function sketched at the right is not. A function that is increasing (or decreasing) on an interval *I* is one-to-one on that interval.

**A6.** If *f* is one-to-one with domain *X* and range *Y*, then there is a function *f* ^{−1}, with domain *Y* and range *X*, such that

*f* ^{−1}(*y*_{0}) = *x*_{0} if and only if *f* (*x*_{0}) = *y*_{0}.

The function *f* ^{−1} is the *inverse* of *f*. It can be shown that *f* ^{−1} is also one-to-one and that its inverse is *f*. The graphs of a function and its inverse are symmetric with respect to the line *y* = *x*.

To find the inverse of *y* = *f* (*x*),

interchange *x* and *y*,

then solve for *y*.

**EXAMPLE 6**

Find the inverse of the one-to-one function *f* (*x*) = *x*^{3} − 1.

**SOLUTION:**

**FIGURE N1–2**

Note that the graphs of *f* and *f* ^{−1} in Figure N1–2 are mirror images, with the line *y* = *x* as the mirror.

**A7.** The *zeros* of a function *f* are the values of *x* for which *f* (*x*) = 0; they are the *x*-intercepts of the graph of *y* = *f* (*x*).

**EXAMPLE 7**

Find zeros of *f* (*x*) = *x*^{4} − 2*x*^{2}.

**SOLUTION:** The zeros are the *x*’s for which *x*^{4} − 2*x*^{2} = 0. The function has three zeros, since *x*^{4} − 2*x*^{2} = *x*^{2} (*x*^{2} − 2) equals zero if *x* = 0, , or