Intro to Mathematics for Computer Science
(06-20415 2012/13 Term 1)
School of CS, University of Birmingham
In-Class Exercises -- 15. Nov 2012
1) You can use books and a calculator.
2) You can talk with your neighbor about the topics.
3) You can ask Dr. Bernd Bohnet or Prof. John Barden for help.
4) Use a extra sheet of paper to solve the exercises of one page.
Ex. 1: The Cartesian product of two sets X and Y, denoted X × Y, is the set of all possible
ordered pairs:
X  Y = { (x,y) | x  Xy Y }
a) What is the Cartesian product of the sets X = {1,2} and Y={a,b,c}? Please, write the
Cartesian product as the set of pairs : X  Y = {(1,a), ...
b) A function is a subset of the Cartesian product f X  Y and written as
f : XY . Underline the correct term (more than one term might be correct):
X
Y
f(x)
x
y
f(X)
range, independent, domain, co-domain, dependent, function, image
range, independent, domain, co-domain, dependent, function, image
range, independent, domain, co-domain, dependent, function, image
range, independent, domain, co-domain, dependent, function, image
range, independent, domain, co-domain, dependent, function, image
range, independent, domain, co-domain, dependent, function, image
c) Given the function f, g and h: XY and X = {1,2,3} and Y={a,b,c} with
f := { (1,a), (2, b)}, g :={(1,a),(2,b),(3,c)}, and h:={(1,a),(2,b),(3,b)}.
Is f, g and h surjective, injective and/or bijective? Does anyone of the functions f, g or
h have an inverse function, if yes define the inverse function?
Ex. 2: Given the equation: x² + y² = 1
a) Transpose the equation to make y the subject.
b) Construct ordered pairs of numbers corresponding to the integer values of x where
-1x1 in intervals of 0.2.
c) Plot the ordered pairs of numbers on a Cartesian graph and join the points with a
continuous curve.
Ex. 3: Given the functions: 2x, x³ - x, sin x, 1/x, 1/(x²-1) and x/x
a) Draft each of the functions. (You do not need to draw the exact graphs.)
b) What are the domain and co-domain of the functions.
Ex.4: Which of the following equations is a function?
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(a) 𝑦 = 𝑥 2 , (b) 𝑦 = 𝑥 2 + 𝑥 + 1 , (c) y = 𝑦 = (√𝑥)3
Ex. 5: Given the two functions f and g by
𝑓(𝑥) = 2𝑥 − 2
for -4<x<4
𝑔(𝑥) = 5𝑥 + 1
for -4 <x<0
a) find the domain and range of h(x) = f(x) - g(x)
b) Calculate the function k(x) = f(x) / g(x)
c) Find the domain of k(x)
Ex. 6: Given the two functions f and g by
2𝑥
𝑓(𝑥) = (5𝑥−10)(3𝑥+3)
𝑔(𝑥) =
1
3𝑥
for -4<x<1 , x  -1
for -2 <x < 0
d) Find the domain and range of h(x) = f(x) - g(x)
e) Calculate the function k(x) = f(x) / g(x)
f) Find the domain of k(x)
Ex. 7: Composition of functions
𝑓(𝑥) = 3x + 12
𝑔(𝑥) = 2𝑥 + 6
a) What is the result of f(g(2)) ?
b) Find each of the expressions f(g(x)) and g(f(x))
Ex. 8: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. 9:Composition of functions
𝑓(𝑥) = sin x
𝑔(𝑥) = cos 𝑥
ℎ(𝑎, 𝑏) = a2 + b2
e) Find the expression of h(f(x),g(x)).
f) Compute h(f(x),g(x)) for x = 1, x=, x=2
g) What is the range of the function?
Ex. 10: Inverse function
a) What happens with the ordered pairs (x,y) of a inverse function?
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b) What is the inverse function of f(x)=x and 𝑔(𝑥) = 𝑥 ?
c) Does 𝑦 = 𝑥 3 have an inverse function and if yes what is the inverse function?
1
d) Does 𝑦 = 𝑥 5 have an inverse function and if yes what is the inverse function?