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

114 Chapter 2 Engineering functions
15 Describe the interval onthexaxisdefined by
(a) |x| < 1.5 (b) |x| > 2
(c) |x+3|<7 (d) |2x|6
(e) |2x−1|5
16 Sketch
(a) f=|−2t| −3t3
(b) f=−|2t| −3t3
(c) 2δ(t)−δ(t −1)+3δ(t +1)
Solutions
1 (a) Multiply input by 7, then subtract2
(b) Square the input, thensubtract 2
(c) Calculate e x ,wherexisthe input, multiply by3
and thenadd 4
(d) Multiply by2, calculate exponential, subtract1
and thendivide by 2
(e) Cube input, addto twice the input and add 5
2 (a) Domain (−∞, ∞)range (−∞, ∞)
(b) Domain [0,5]range [−2,23]
(c) Domain [0,2]range [7,3e 2 +4]
(d) Domain [0,∞) range [0,∞)
(e) Domain [−2,2]range [−7,17]
3 (a) y −1 (x) = x 2
(b) f −1 (t) = t +3
8
(c) h −1 (x) = 3 2 (x−1)
(d) m −1 (r) = 1 −r
3
(e) H −1 (s) = 3
s −2
(f) f −1 (v) =e v
(g) f −1 (t) = 1 2 lnt
(h) g −1 (v) = e v−1
(i) g −1 (v) =e v (
−1
)
(j) y −1 t
(t) = ln +2
3
4 (a) e 2t (b) e x (c) e λ (d) e t−λ
5 (a) ln(λ 2 +1) (b) ln((t − λ) 2 +1)
8 (a) t 2
(b) (t+3) 2 −3ort 2 +6t+6
(c) 3(t +3)
(d) 3(t 2 −3)
(e) 3t 2
(f) (3t+3) 2 −3or9t 2 +18t+6
10 (a) e 5x (b) 1 (c) e 8
(d) e 2x (e) e 4x (f) ln6
(g) lnt 7 (h) x (i) x
(j) x 2 (k) x (l) 2x
(m) x 2 (n) 3 +2x
11 (a) 1.3246 (b) 3.6702
(c) −1.7918 (d) ±2.0234
(e) −1.3863
12 (a) 11.0232 (b) 0.6851
(c) 8.3891 (d) ±5.7116
(e) 23.6243 (f) 44.5370
13 (a) 10coshx +4sinhx
(b) coshx +11sinhx
coshx +5sinhx
(c)
2
1
(d)
2coshx
coshx +sinhx
(e)
1+coshx+sinhx
7
14 (a)
2 ex + 3 2 e−x
e x −e −x +2
(b)
e x +e −x
5e x +11e −x
(c)
8
−2
(d)
e x +3e −x
( )
e x −e −x 2
(e)
2
15 (a) −1.5 <x<1.5
(b) x>2andx<−2
(c) −10<x<4
(d) x3andx−3
(e) −2x3