introduction to financial mathematics worked examples functions
TRANSCRIPT
Introduction to Financial Mathematics
Worked Examples
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 followingpages. Click on the link with each question to go straight to the relevant page. You willneed to have the question handy to refer to while watching the videos.
Questions
1. See Page 3 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}
2. See Page 5 for worked solutions.For each of the following quadratic equations find the y-intercept, the axis of symmetry,the vertex and the zeros and sketch the function, showing each of these features onyour graph. (a) y = 3x2 + 2x− 8 (b) y = 2x2 + 4x+ 5
3. See Page 7 for worked solutions.Find the y-intercept, symmetry point and zeros of the cubic equationy = x3 − x2 + x− 1 = (x− 1)(x2 + 1). Sketch the graph of this function.
4. See Page 8 for worked solutions.Give the domain and range of the rational functions
f(x) =1
x+ 2and g(x) =
1
x2 + 1.
5. See Page 9 for worked solutions.A government assesses income tax T on each citizen’s income Y as follows: for incomeup to $10,000, no tax; a tax rate of 2% on income from $10,000 to $50,000 and 4% onincome above $50,000 up to a maximum tax of $2,800. Sketch T as a function of Y .
6. See Page 10 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
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.
(a) f + g (b) g/f (c) f ◦ g (d) g ◦ f(e) f · g (f) h · g (g) f ◦ h (h) h ◦ f .
8. See Page 16 for worked solutions.Sketch the functions g(t) = 2t + 2−t and h(t) = 2t − 2−t.
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.Find the inverse of the function g(x) = x3 + 2, and plot both g and g−1 on the sameaxes.
11. See Page 20 for worked solutions.Evaluate the following limits (if possible):
(a) limx→0
((x3 + 2) sin(π/2− x)
)(b) lim
x→0
(1
x− 2
x2
)(c) lim
x→a
√x−√a
x− a(d) lim
x→ 2
x2 − 3x− 4
x+ 2(Note that the function sin is not part of the IFM course, so you can ignore part (a).)
2
1. Click here to go to question list.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}
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 yourpage!
Click here to see video of this example on YouTube.
3
4
2. Click here to go to question list.For each of the following quadratic equations find the y-intercept, the axis of symmetry,the vertex and the zeros and sketch the function, showing each of these features onyour graph. (a) y = 3x2 + 2x− 8 (b) y = 2x2 + 4x+ 5
Click here to see video of this example on YouTube.
5
6
3. Click here to go to question list.Find the y-intercept, symmetry point and zeros of the cubic equationy = x3 − x2 + x− 1 = (x− 1)(x2 + 1). Sketch the graph of this function.
Click here to see video of this example on YouTube.
7
4. Click here to go to question list.Give the domain and range of the rational functions
f(x) =1
x+ 2and g(x) =
1
x2 + 1.
Click here to see video of this example on YouTube.
8
5. Click here to go to question list.A government assesses income tax T on each citizen’s income Y as follows: for incomeup to $10,000, no tax; a tax rate of 2% on income from $10,000 to $50,000 and 4% onincome above $50,000 up to a maximum tax of $2,800. Sketch T as a function of Y .
Click here to see video of this example on YouTube.
9
6. 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.
Click here to see video of this example on YouTube.
10
11
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 .
Click here to see video of this example on YouTube.
12
13
14
15
8. Click here to go to question list.Sketch the functions g(t) = 2t + 2−t and h(t) = 2t − 2−t.
Click here to see video of this example on YouTube.
16
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.
Click here to see video of this example on YouTube.
17
18
10. Click here to go to question list.Find the inverse of the function g(x) = x3 + 2, and plot both g and g−1 on the sameaxes.
Click here to see video of this example on YouTube.
19
11. Click here to go to question list.Evaluate the following limits (if possible):
(a) limx→0
((x3 + 2) sin(π/2− x)
)(b) lim
x→0
(1
x− 2
x2
)(c) lim
x→a
√x−√a
x− a(d) lim
x→ 2
x2 − 3x− 4
x+ 2(Note that the function sin is not part of the IFM course, so you can ignore part (a).)
Click here to see video of this example on YouTube.
20
21