James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 1.2 Mathematical Models: A Catalog of Essential Functions 33
y
y=log x
y=log£ x
1
0 1
y=log∞ x
figure 21
x
y=log¡¸ x
21 shows the graphs of four logarithmic functions with various bases. In each case the
domain is s0, `d, the range is s2`, `d, and the function increases slowly when x . 1.
ExamplE 6 Classify the following functions as one of the types of functions that we
have discussed.
(a) f sxd − 5 x (b) tsxd − x 5
(c) hsxd −
1 1 x
(d) ustd − 1 2 t 1 5t 4
1 2 sx
SOLUTION
(a) f sxd − 5 x is an exponential function. (The x is the exponent.)
(b) tsxd − x 5 is a power function. (The x is the base.) We could also consider it to be a
polynomial of degree 5.
(c) hsxd −
1 1 x is an algebraic function.
1 2 sx
(d) ustd − 1 2 t 1 5t 4 is a polynomial of degree 4.
■
1. 2 Exercises
1–2 Classify each function as a power function, root function,
polynomial (state its degree), rational function, algebraic function,
trigonometric function, exponential function, or logarithmic function.
1. (a) f sxd − log 2 x (b) tsxd − s 4 x
(c) hsxd − 2x 3
2
1 2 x
2
(d) ustd − 1 2 1.1t 1 2.54t
(e) vstd − 5 t
(f) wsd − sin cos 2
2. (a) y − x (b) y − x
(c) y − x 2 s2 2 x 3 d
(e) y −
s
1 1 s
(d) y − tan t 2 cos t
(f) y − sx 3 2 1
1 1 s 3 x
3–4 Match each equation with its graph. Explain your choices.
(Don’t use a computer or graphing calculator.)
3. (a) y − x 2 (b) y − x 5 (c) y − x 8
f
y
0
g
h
x
4. (a) y − 3x (b) y − 3 x (c) y − x 3 (d) y − s 3 x
f
5–6 Find the domain of the function.
5. f sxd −
cos x
1 2 sin x
y
G
F
6.
g
tsxd −
x
1
1 2 tan x
7. (a) Find an equation for the family of linear functions with
slope 2 and sketch several members of the family.
(b) Find an equation for the family of linear functions such
that f s2d − 1 and sketch several members of the family.
(c) Which function belongs to both families?
8. What do all members of the family of linear functions
f sxd − 1 1 msx 1 3d have in common? Sketch several
members of the family.
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