082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.3 Basic concepts of functions 69
14 Sketch
f(t)=
{
t 0t<2
5−2t 2t<3
Isthe function piecewise continuous orcontinuous?
State,ifthey exist, the position ofany discontinuities.
15 Thefunctionh(t)is defined by
{
2 −t 0t<2
h(t) =
2t−4 2t3
andh(t)has period 3.Sketchh(t)onthe interval
[0,6].
16 The functiong(t)is defined by
{
1 0 t1
g(t) =
2−t 1<t<2
andg(t)hasperiod 2.Sketchg(t)onthe interval
[−1,4].State any pointsofdiscontinuity.
Solutions
2 (a) Square the input and thenmultiply by2; domain
(−∞, ∞),range [0, ∞)
(b) Square the input, then subtract1; domain [0,∞),
range [−1,∞)
(c) Multiply input by3and subtract4; domain
[0, ∞),range [−4,∞)
(d) Cube the input;domain (−∞, ∞),range
(−∞,∞)
(e) Multiply input by0.5and then add2; domain
[−2,10], range [1,7]
(f) Multiply input by3and then subtract2; domain
[3,8],range [7,22]
3 (a) 19 (b) −11 (c) 5α +4
(d) 5x+9 (e) 15α+4 (f) 5x 2 +4
4 (a) −4 (b) 16 (c) 41
(d) 5x 2 −4 (e) 20t 2 −20t +1
5 3183ohms
8 (a) 2(t −1) (b) 2t 2 (c) t 2 −1
(d) 2t −1 (e) (t −1) 2 (f) 4t 2
(g) 4t (h) t −2 (i) t 4
(j) 2(t 2 −1) (k) 2t 2 −1 (l) (2t −1) 2
9 (a) 9t 2 1
+12t +5 (b)
t 2 +1
3
(c)
t +2 (d) 1
t 2 +1
9
(e)
t 2 + 12 +5
t
t
10 (a) (b) t −1
2 2
x +2 2
11 (a) (b)
3 x
12 (a) 6t +3 (b) t −3
2
(d) t −3 (e) t −3
6 6
13 SeeFigureS.1.
1 −t
(c)
3
1
(c)
x −1
t
(c)
3
6 (a) many-to-one (b) one-to-one
(c) one-to-one
(e) one-to-one
7 (a)f −1 (x)=x−4
(b) g −1 (t) = t −1
3
(c) y −1 (x) =x 1/3
(d) h −1 (t) =3t +8
(e) f −1 (t)=3t+1
(f) h −1 (x) = (x +1) 1/3
(g) k −1 (v) =7−v
(h) m −1 (n) = 1−3n
2
(d) one-to-one
(f) one-to-one
f
3
2
1
(a)
0
a
1
1
2 3 4 t
(c)
0 1 2 t
FigureS.1
(b)
(d)
g
2
1
0 1 2 3 x
b
2
1
0 1 2 3 x