Mathematics IAWorked 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 onthe following pages. Click on the link with each question to go straight tothe relevant page.

Questions

1. See Page 3 for worked solutions.Suppose x is an irrational number and y is any real number. Show thatat 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 ofintegers:

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 | y2 = 9}(c) {x ∈ R | x2 ≥ 1} ∩ {x ∈ R | x2 < 4}

4. See Page 7 for worked solutions.

(a) Let f(x) =

−1− x if x < −1

0 if − 1 ≤ x < 0

x2 − 4 if 1 ≤ x ≤ 2

.

Sketch a graph of f and write down the domain and range of f .

(b) Consider the real-valued function g(x) =

√x+ 4

(x2 + 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 = 12

5. Find the exact

value of (a) sin θ (b) sin 2θ (c) sin θ2

.

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 integerfunction. It is also written as b∗c.)

7. See Page 12 for worked solutions.

Let f(x) = x2 − 4, g(x) =√x and h(x) =

{0 if x ≤ 2

x− 2 if x > 2. Find

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.

Consider the function f(x) =2x+ 1

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) arctanx = −π

3(b) tan(arcsinx) = 2.

11. See Page 20 for worked solutions.Solve for x ∈ R: (a) 4x = 23x+2 (b) 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 thatat 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 ofintegers:

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 | y2 = 9}(c) {x ∈ R | x2 ≥ 1} ∩ {x ∈ R | x2 < 4}

NOTE: The following solutions have an error in part (c): the solutionfinds 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 whencopying the question onto your page!

Link to video on YouTube

4. Click here to go to question list.

(a) Let f(x) =

−1− x if x < −1

0 if − 1 ≤ x < 0

x2 − 4 if 1 ≤ x ≤ 2

.

Sketch a graph of f and write down the domain and range of f .

(b) Consider the real-valued function g(x) =

√x+ 4

(x2 + 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 = 12

5. Find the exact

value of (a) sin θ (b) sin 2θ (c) sin θ2

.

Link to video on YouTube

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 integerfunction. It is also written as b∗c.)Link to video on YouTube

7. Click here to go to question list.

Let f(x) = x2 − 4, g(x) =√x and h(x) =

{0 if x ≤ 2

x− 2 if x > 2. Find

expressions for the following functions and give the domain in each case.

(a) f + g (b) g/f (c) f ◦ g (d) g ◦ f(e) f · g (f) h · g (g) f ◦ h (h) h ◦ f .

Link to video on YouTube

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.

Consider the function f(x) =2x+ 1

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) arctanx = −π

3(b) tan(arcsinx) = 2.

Link to video on YouTube

11. Click here to go to question list.Solve for x ∈ R: (a) 4x = 23x+2 (b) log8(x+ 2) + log8(x) = 1.

Link to video on YouTube