Intro to Mathematics for Computer Science
(06-20415 2012/13 Term 1)
School of CS, University of Birmingham
In-Class Exercises -- 23. Nov 2012
Ex. 1: Given that a(x) =5x, b(x)=x4, c(x)=x+3 and d(x) = √𝑥 find:
(a) f(x) = a[b(c[d(x)])]
(b) f(x)=a(a[d(x)])
(c) f(x) = b[c(b[c(x)])]
Ex. 2:Trigometric functions
a)
b)
c)
d)
What is the radius of the unit circle r and what is its perimeter?
Assign the appropriate function to a, b and d (sin ϴ, cos ϴ, tan ϴ)?
Draw the graphs of sin ϴ, cos ϴ, and tan ϴ.
What is a² + b² = ?
Ex. 3: Use your calculator
a) sin (-5),
b) cos (45°) and
c) tan (-113°)
Ex. 4: Find the period and amplitude of each of the following functions:
a) 4 cos (5 ϴ) , 0.5 cos ( 7 ϴ ) , cos ( ϴ / 2 + 𝜋/3 )
Ex. 3 Composition of functions
𝑓(𝑥) = sin x
𝑔(𝑥) = cos 𝑥
ℎ(𝑎, 𝑏) = a2 + b2
a) Find the expression of h(f(x),g(x)).
b) Compute h(f(x),g(x)) for x = 1, x=, x=2
c) What is the range of the function?
Ex. 4: Given are the definitions and the functions below
surjectiv
<=>
f(A) = B
injectiv
<=>
x1 ≠ x2 => f(x1) ≠ f(x2)
bijectiv
<=>
f is surjective and injectiv
g: A ⟶
B
k: X
a
⟶
Y
j:X
0
1
i
2
ii
b
i
1
1
c
ii
2
4
⟶
B
What is true and why?
1: a) g is surjectiv and injectiv. b) g is not surjectiv but injective. c) g is surjective but not injectiv.
2: a)k is surjective and injective but not bijectiv. b) k is bijective c) g is bjectiv and not surjectiv.
3: a) j is surjectiv. b) j is injectiv. c) j is not a function. d) j is bijectiv. e) surjectiv and injectiv but not
bijectiv.
4: a) g has an inverse function. b) g and k have an inverse function. c) Only j has an inverse function.
5) Why is 𝑦 = ±√𝑥 3 not a function.
Ex. 5 Calculate the inverse function of
a) y=-7x
b) 𝑦 = √𝑥 3