Calculus 2nd Edition Rogawski

22 CHAPTER 1 PRECALCULUS REVIEW
The domain of f (x) is the set of numbers x such that Q(x) ̸= 0. For example,
f (x) = 1 x 2 domain {x : x ̸= 0}
−2
FIGURE 3 The algebraic function
f (x) = √ 1 + 3x 2 − x 4 .
y
Any function that is not algebraic is called
transcendental. Exponential and
trigonometric functions are examples, as
are the Bessel and gamma functions that
appear in engineering and statistics. The
term “transcendental” goes back to the
1670s, when it was used by Gottfried
Wilhelm Leibniz (1646–1716) to describe
functions of this type.
2
x
h(t) = 7t6 + t 3 − 3t − 1
t 2 − 1
domain {t : t ̸= ±1}
Every polynomial is also a rational function [with Q(x) = 1].
• An algebraic function is produced by taking sums, products, and quotients of roots
of polynomials and rational functions (Figure 3):
f (x) =
√1 + 3x 2 − x 4 , g(t) = ( √ t − 2) −2 , h(z) = z + z−5/3
5z 3 − √ z
A number x belongs to the domain of f if each term in the formula is defined and
the result does not involve division by zero. For example, g(t) is defined if t ≥ 0
and √ t ̸= 2, so the domain of g(t) is D ={t : t ≥ 0 and t ̸= 4}. More generally,
algebraic functions are defined by polynomial equations between x and y. In this
case, we say that y is implicitly defined as a function of x. For example, the equation
y 4 + 2x 2 y + x 4 = 1 defines y implicitly as a function of x.
• Exponential functions: The function f (x) = b x , where b>0, is called the exponential
function with base b. Some examples are
( ) 1 x
f (x) = 2 x , g(t) = 10 t , h(x) = , p(t) = ( √ 5) t
3
Exponential functions and their inverses, the logarithmic functions, are treated in
greater detail in Chapter 7.
• Trigonometric functions are functions built from sin x and cos x. These functions
are discussed in the next section.
Constructing New Functions
Given functions f and g, we can construct new functions by forming the sum, difference,
product, and quotient functions:
(f + g)(x) = f (x) + g(x), (f − g)(x) = f (x) − g(x)
( ) f
(fg)(x) = f (x) g(x),
(x) = f (x) (where g(x) ̸= 0)
g g(x)
For example, if f (x) = x 2 and g(x) = sin x, then
(f + g)(x) = x 2 + sin x, (f − g)(x) = x 2 − sin x
(fg)(x) = x 2 sin x,
( f
g
)(x) = x2
sin x
We can also multiply functions by constants. A function of the form
c 1 f (x) + c 2 g(x)
(c 1 ,c 2 constants)
is called a linear combination of f (x) and g(x).