Solução_Calculo_Stewart_6e

F.
TX.10
SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ¤ 13
61. f is an odd function because its graph is symmetric about the origin. g is an even function because its graph is symmetric with
respect to the y-axis.
63. (a) Because an even function is symmetric with respect to the y-axis, and the point (5, 3) is on the graph of this even function,
the point (−5, 3) must also be on its graph.
(b) Because an odd function is symmetric with respect to the origin, and the point (5, 3) is on the graph of this odd function,
the point (−5, −3) must also be on its graph.
65. f(x) = x
x 2 +1 .
f(−x) =
−x
(−x) 2 +1 =
So f is an odd function.
−x
x 2 +1 = − x
x 2 +1 = −f(x).
67. f(x) = x
x +1 ,sof(−x) =
−x
−x +1 =
x
x − 1 .
Sincethisisneitherf(x) nor −f(x), the function f is
neither even nor odd.
69. f(x) =1+3x 2 − x 4 .
f(−x) =1+3(−x) 2 − (−x) 4 =1+3x 2 − x 4 = f(x).
So f is an even function.
1.2 Mathematical Models: A Catalog of Essential Functions
1. (a) f(x) = 5√ x is a root function with n =5.
(b) g(x) = √ 1 − x 2 is an algebraic function because it is a root of a polynomial.
(c) h(x) =x 9 + x 4 is a polynomial of degree 9.
(d) r(x) = x2 +1
is a rational function because it is a ratio of polynomials.
x 3 + x
(e) s(x) =tan2x is a trigonometric function.
(f ) t(x) =log 10 x is a logarithmic function.
3. We notice from the figure that g and h are even functions (symmetric with respect to the y-axis) and that f is an odd function
(symmetric with respect to the origin). So (b) y = x 5 must be f. Sinceg is flatter than h near the origin, we must have
(c) y = x 8 matched with g and (a) y = x 2 matched with h.