Mathematics IA
Worked Examples
CALCULUS: FUNCTIONS
Produced by the Maths Learning Centre,
The University of Adelaide.
May 3, 2013
The questions on this page have worked solutions and links to videos on
the following pages. Click on the link with each question to go straight to
the relevant page.
Questions
1. See Page 3 for worked solutions.
Suppose x is an irrational number and y is any real number. Show that
at least one of x + y or x − y is irrational.
2. See Page 4 for worked solutions.
Show that the following number is rational by writing it as a fraction of
integers:
5.013590590590590 . . .
3. See Page 5 for worked solutions.
Write the following sets in interval notation:
(a) {x ∈ R | − 1 ≤ x < 6} ∪ {a2 + 1 | a ∈ R}
(b) {x ∈ R | 2x < 5} \ {y ∈ R | y 2 = 9}
(c) {x ∈ R | x2 ≥ 1} ∩ {x ∈ R | x2 < 4}
4. See Page 7 for worked solutions.


−1 − x if x < −1
(a) Let f (x) = 0
if − 1 ≤ x < 0 .

 2
x −4
if 1 ≤ x ≤ 2
Sketch a graph of f and write down the domain and range of f .
√
x+4
.
(b) Consider the real-valued function g(x) = 2
(x + 1)(x2 − 1)
Find the largest possible domain for g.
1
5. See Page 9 for worked solutions.
Let θ be an angle between π and 2π such that tan θ =
value of (a) sin θ (b) sin 2θ (c) sin 2θ .
12
.
5
Find the exact
6. See Page 11 for worked solutions.
Solve for x ∈ R: (a) J2x − 3K = 5 (b) |x − 5| = |3 − 2x|
(Note that I have used the notation J∗K for the floor or greatest integer
function. It is also written as b∗c.)
7. See Page 12 for worked solutions.
(
√
0
if x ≤ 2
. Find
Let f (x) = x2 − 4, g(x) = x and h(x) =
x − 2 if x > 2
expressions for the following functions and give the domain in each case.
8. See Page 16 for worked solutions.
Suppose that f and g are both odd functions.
(a) Is the function f + g odd or even?
(b) Is the function f · g odd or even?
9. See Page 17 for worked solutions.
2x + 1
.
Consider the function f (x) =
x−1
(a) State the domain and range of f .
(b) Show that f is a 1–1 function.
(c) Find the inverse function f −1 .
10. See Page 19 for worked solutions.
Solve for x ∈ R: (a) arctan x = −π
3
11. See Page 20 for worked solutions.
Solve for x ∈ R: (a) 4x = 23x+2
(b)
(b)
tan(arcsin x) = 2.
log8 (x + 2) + log8 (x) = 1.
1. Click here to go to question list.
Suppose x is an irrational number and y is any real number. Show that
at least one of x + y or x − y is irrational.
Link to video on YouTube
2. Click here to go to question list.
Show that the following number is rational by writing it as a fraction of
integers:
5.013590590590590 . . .
Link to video on YouTube
3. Click here to go to question list.
Write the following sets in interval notation:
(a) {x ∈ R | − 1 ≤ x < 6} ∪ {a2 + 1 | a ∈ R}e
(b) {x ∈ R | 2x < 5} \ {y ∈ R | y 2 = 9}
(c) {x ∈ R | x2 ≥ 1} ∩ {x ∈ R | x2 < 4}
NOTE: The following solutions have an error in part (c): the solution
finds the set {x ∈ R | x2 > 1} ∩ {x ∈ R | x2 < 4} – that is, with “x2 > 1”
instead of “x2 ≥ 1”. It just goes to show you have to be very carful when
copying the question onto your page!
Link to video on YouTube
4. Click here to go to question list.


−1 − x if x < −1
(a) Let f (x) = 0
if − 1 ≤ x < 0 .

 2
x −4
if 1 ≤ x ≤ 2
Sketch a graph of f and write down the domain and range of f .
√
x+4
(b) Consider the real-valued function g(x) = 2
.
(x + 1)(x2 − 1)
Find the largest possible domain for g.
Link to video on YouTube
5. Click here to go to question list.
Let θ be an angle between π and 2π such that tan θ =
value of (a) sin θ (b) sin 2θ (c) sin 2θ .
Link to video on YouTube
12
.
5
Find the exact
6. Click here to go to question list.
Solve for x ∈ R: (a) J2x − 3K = 5 (b) |x − 5| = |3 − 2x|
(Note that I have used the notation J∗K for the floor or greatest integer
function. It is also written as b∗c.)
Link to video on YouTube
7. Click here to go to question list.
(
√
0
if x ≤ 2
Let f (x) = x2 − 4, g(x) = x and h(x) =
. Find
x − 2 if x > 2
expressions for the following functions and give the domain in each case.
(a) f + g
(e) f · g
(b) g/f
(f) h · g
Link to video on YouTube
(c) f ◦ g
(g) f ◦ h
(d) g ◦ f
(h) h ◦ f .
8. Click here to go to question list.
Suppose that f and g are both odd functions.
(a) Is the function f + g odd or even?
(b) Is the function f · g odd or even?
Link to video on YouTube
9. Click here to go to question list.
2x + 1
.
Consider the function f (x) =
x−1
(a) State the domain and range of f .
(b) Show that f is a 1–1 function.
(c) Find the inverse function f −1 .
Link to video on YouTube
10. Click here to go to question list.
Solve for x ∈ R: (a) arctan x =
Link to video on YouTube
−π
3
(b)
tan(arcsin x) = 2.
11. Click here to go to question list.
Solve for x ∈ R: (a) 4x = 23x+2
Link to video on YouTube
(b)
log8 (x + 2) + log8 (x) = 1.