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

68 Chapter 2 Engineering functions
EXERCISES2.3
1 Representthe followingintervals onthe real line:
(a) [1,3] (b) [2,4)
(c) (0,3.5) (d) [−2,0)
(e) (−1,1] (f) 2x<4
(g) 0<x<2 (h) −3x −1
(i) 0x<3
2 Describe the rule associated with the following
functions,sketch their graphs and state their domains
and ranges:
(a) f(x) =2x 2
(b)f(x)=x 2 −1
(c)g(t)=3t−4
(d)y(x) =x 3
0x
0t
(e)f(t)=0.5t+2 −2t10
(f)z(x)=3x−2
3 Iff(x)=5x+4,find
(a) f(3)
(b) f(−3)
(c) f(α)
(d) f(x+1)
(e) f(3α)
(f) f(x 2 )
4 Ifg(t) =5t 2 −4,find
(a) g(0)
(b) g(2)
(c) g(−3)
(d) g(x)
(e) g(2t −1)
3x8
5 Thereactance,X C ,offered by acapacitor is given by
X C = 1 ,where f isthe frequency ofthe applied
2πfC
alternating current, andC is the capacitance ofthe
capacitor. IfC = 10 −6 F,findX C when f = 50 Hz.
6 Classifythe functions in Question2asone-to-oneor
many-to-one.
7 Find the inverse ofthe followingfunctions:
(a) f(x)=x+4
(b) g(t) =3t +1
(c) y(x) =x 3
(d) h(t) = t −8
3
(e) f(t) = t −1
3
(f) h(x) =x 3 −1
(g) k(v)=7−v
(h) m(n) = 3 1 (1 −2n)
8 Givenf(t) =2t,g(t) =t −1andh(t) =t 2 write
expressions for
(a) f(g(t)) (b) f(h(t))
(c) g(h(t)) (d) g(f(t))
(e) h(g(t)) (f) h(f(t))
(g) f(f(t)) (h) g(g(t))
(i) h(h(t)) (j) f(g(h(t)))
(k) g(f(h(t))) (l) h(g(f(t)))
9 Givenf(t) =t 2 +1,g(t) =3t +2andh(t) = 1 t ,
write expressions for
(a) f(g(t)) (b) f(h(t))
(c) g(h(t)) (d) h(f(t))
(e) f(g(h(t)))
10 Given f(t) =2t,g(t) =2t +1,h(t) =1−3t,write
expressions forthe following:
(a) f −1 (t) (b)g −1 (t) (c)h −1 (t)
11 Givena(x) = 3x −2,b(x) = 2 x ,c(x)=1+1 x write
expressions for
(a)a −1 (x) (b)b −1 (x) (c)c −1 (x)
12 Givenf(t) =2t +3,g(t) =3tandh(t) = f(g(t))
write expressions for
(a) h(t)
(b) f −1 (t)
(c) g −1 (t)
(d) h −1 (t)
(e) g −1 (f −1 (t))
Whatdoyou noticeabout(d)and(e)?
13 Sketch the following functions:
{
t 0t3
(a) f(t) =
3 3<t4
{
2 −x 0x<1
(b) g(x) =
2 1x3
{
1 −t 0t1
(c) a(t) =
t−1 1<t2
⎧
⎪⎨
2 0 x1
(d) b(x) = 1 1<x2
⎪⎩
3−x 2<x3