Solução_Calculo_Stewart_6e

F.
TX.10
36 ¤ CHAPTER 1 FUNCTIONS AND MODELS
3. f(x) =x 2 − 2x +3,sof(a + h) =(a + h) 2 − 2(a + h)+3=a 2 +2ah + h 2 − 2a − 2h +3,and
f(a + h) − f(a)
h
= (a2 +2ah + h 2 − 2a − 2h +3)− (a 2 − 2a +3)
h
=
h(2a + h − 2)
h
=2a + h − 2.
5. f(x) =2/(3x − 1). Domain: 3x − 1 6= 0 ⇒ 3x 6= 1 ⇒ x 6= 1 . D =
−∞, 1 3 3 ∪ 1 , ∞ 3
Range: all reals except 0 (y =0is the horizontal asymptote for f.) R =(−∞, 0) ∪ (0, ∞)
7. h(x) =ln(x +6). Domain: x +6> 0 ⇒ x>−6. D =(−6, ∞)
Range: x +6> 0,soln(x +6)takes on all real numbers and, hence, the range is R.
R =(−∞, ∞)
9. (a)Toobtainthegraphofy = f(x)+8,weshiftthegraphofy = f(x) up 8 units.
(b)Toobtainthegraphofy = f(x +8),weshiftthegraphofy = f(x) left 8 units.
(c) To obtain the graph of y =1+2f(x), we stretch the graph of y = f(x) vertically by a factor of 2, and then shift the
resulting graph 1 unit upward.
(d)Toobtainthegraphofy = f(x − 2) − 2, we shift the graph of y = f(x) right 2 units (for the “−2” insidethe
parentheses), and then shift the resulting graph 2 units downward.
(e)Toobtainthegraphofy = −f(x),wereflect the graph of y = f(x) about the x-axis.
(f)Toobtainthegraphofy = f −1 (x),wereflect the graph of y = f(x) about the line y = x (assuming f is one–to-one).
11. y = − sin 2x: Start with the graph of y =sinx, compress horizontally by a factor of 2,andreflect about the x-axis.
13. y = 1 (1 + 2 ex ):
Startwiththegraphofy = e x ,
shift 1 unit upward, and compress
vertically by a factor of 2.
15. f(x) = 1
x +2 :
Startwiththegraphoff(x) =1/x
and shift 2 units to the left.