(1/30)
MA160/MA161 : Semester 1 Calculus
Week 3: Functions (domain, range, piecewise,
odd and even)
http://www.maths.nuigalway.ie/~niall/MA161-1
22nd and 23rd of September, 2014
Tutorials
(2/30)
MA160 Tutorials started last week:
Thursday at 13.00 in IT125G.
.....................................................................
MA161 Tutorials start from today (19th September). For those
registered for MA161 should attend one of:
1
Monday at 14:00 in IT203.
2
Tuesday at 09:00 in AM105
3
Tuesday at 11:00 in IT206
4
Tuesday at 18:00 in AC201
5
Wednesday at 18:00 in AM104
6
Friday at 11:00 in IT203
Assignments
(3/30)
MA161 Set “Problem Set 1”
Deadline: 5pm, Friday 03 Oct 2014
Corresponds to: Calculus and Algebra from Weeks 1–3.
Number of questions: 15
Max attempts: 10
Access at: Blackboard.
See also: www.maths.nuigalway.ie/∼niall/MA161-1
Credit: 5% of MA161.
.....................................................................
MA160 Set “Problem Sheet 1”
Deadline: 5pm, Friday 26 September, 2014
Corresponds to: Calculus and Algebra from Weeks 1–3.
Number of questions: 10
Max attempts: 10
Access at: http://brnsrv.nuigalway.ie:8128
Credit: 4.5% of MA160 (approximately).
MA161 Problem Set 1
(4/30)
All MA161 homework assignments are run using an on-line system called
WeBWorK. It assigns a unique problem set for each student. Everyone
has the same number and type of exercises, but the numbers/coefficients
are different for every one.
You access the homework by
logging onto Blackboard and
selecting the module
1415-MA161, and then
choosing the assignment:
Outline
1 Recall...
Domain, co-domain and range
2 Piecewise functions
The absolute value function
Other examples
3 Even and Odd functions
4 1. Linear functions
5 2. Polynomials
6 Exercises
(5/30)
Recall...
Domain, co-domain and range (6/30)
Recall that a function is a rule that maps values from one set to another.
In MA16x, we are mainly concerned with functions from R to R.
Given the formula for a function, we’ll frequently have to figure out
What is the domain of f ? That is the set of all numbers x ∈ R,
such that f (x) makes sense;
What is the range of f ? That is the set of all numbers y ∈ R, such
that y = f (x) for some x.
When figuring out the domain, we need to take into account:
for what values of R for which f defined. In particular, we must
avoid dividing by zero. function f (x) = 1/x is not defined at x = 0.
for what values of R does f map to a real number? For example,
√
x = x 1/2 is a real number only when x ≥ 0;
Recall...
Domain, co-domain and range (7/30)
Example
What (subset of R) are the largest possible domain and range for the
√
function f (x) = x + 2?
Recall...
Domain, co-domain and range (8/30)
Example (MA160 Problem Sheet 1, Question 8)
What is the domain of the following real-valued function?
f (x) =
17
.
x 3 + 2x 2 − 8x
Recall...
Domain, co-domain and range (9/30)
Example: Suppose that f is a function f : X → Y given by
f (x) = 1 + x 3 .
1
If the domain X is the set [0, ∞), what this the range?
2
What domain gives the range [0, ∞)?
Piecewise functions
The absolute value function (10/30)
So far, we have met inequality expressions in solving problems like
Let f (x) = x 3 − 4x. For what values of x is f (x) ≥ 0?
However, we will often have functions where inequality expressions are
used in the definition of the function. These are piecewise-defined
functions.
The most important example of this is the absolute value function:
Absolute value
The absolute value of a real number x, denoted as |x| is:
(
−x x < 0
|x| =
x
x ≥ 0.
Piecewise functions
The absolute value function (11/30)
Examples: Sketch the graph of the following functions:
A. f (x) = |x|
B. f (x) = |x − 1|
C. f (x) = |x| − 1
Piecewise functions
Examples: The Heavyside function...
Other examples (12/30)
Piecewise functions
Other examples (13/30)
Example: Sketch the graph of the piecewise defined function
(
1 + x x < −1
f (x) = 1 2
x ≥ −1.
2x
Solution
1
f (x)
0
−1
−2
−3
−3
−2
−1
x
0
1
Piecewise functions
Other examples (14/30)
A piecewise defined function that you might be familiar with is the one
that maps scores in Leaving Cert exams to CAO points. Lets consider the
mapping for higher-level papers: f : [0, 100] → [0, 100]
Grade
A1
A2
B1
B2
B3
C1
C2
C3
D1
D2
D3
E
F
NG
Score
90 ≤ x ≤ 100
85 ≤ x < 90
80 ≤ x < 85
75 ≤ x < 80
70 ≤ x < 75
65 ≤ x < 70
60 ≤ x < 65
55 ≤ x < 60
50 ≤ x < 55
45 ≤ x < 50
40 ≤ x < 45
25 ≤ x < 40
10 ≤ x < 25
0 ≤ x < 10
Points
100
90
85
80
75
70
65
60
55
50
45
0
0
0
Piecewise functions
Other examples (15/30)
Example: Sketch both the functions f (x) = x 3
3
1
0
−1
−2
−3
−2 −1
0
x
1
2
g (x) = |x 3 |.
8
7
6
5
4
3
2
1
0
−1
−2−1 0 1 2
x
g(x) = |x3 |
f (x) = x3
2
and
Piecewise functions
Other examples (16/30)
Example (From MA161 Problem Set 1, Question 2)
Solve the inequality:
Even and Odd functions
Definition
A function f is called even if f (−x) = f (x).
A function f is called odd if f (−x) = −f (x).
It is possible for a function to be neither odd nor even.
Example
Show that the function f (x) = x 2 is even.
(17/30)
Even and Odd functions
Example
Show that the function f (x) =
x3 − x
is odd.
x2 + 1
(18/30)
Even and Odd functions
Example
1
2
Is the function f (x) = |x| odd or even?
Is the function f (x) = |3 − x| odd or even?
(19/30)
Even and Odd functions
(20/30)
The notion of “even” and “odd” functions is closely related to symmetry .
An even function is symmetric about the y -axis, while an odd function is
symmetric about the point (0, 0).
Examples:
3
1
g(x) = |x3 |
f (x) = x3
2
0
−1
−2
−3
−2 −1
8
7
6
5
4
3
2
1
0
−1
−2−1 0 1 2
x
0
x
1
2
Even and Odd functions
(21/30)
Example: (MA160/MA161 Paper 1, 2012/13, Q4(c)) For each of the
following functions, determine if it is odd, even or neither:
1
(i) f (x) = + x 7 + x 9 ;
x
(ii) g (x) = −5x 2 − 3x 4 − 2.
A catalogue of functions
(22/30)
[The following material follows Section 1.2 of Stewart’s Calculus quite
closely .]
We’ll now recap some of the most common functions that you’ll meet
during the course of the year. They include
1. Linear functions;
2. Polynomials;
3. Power functions;
4. Rational functions;
5. Algebraic functions;
6. Trigonometric functions;
7. Exponential functions;
8. Logarithms.
0. Trivial functions
But first we’ll look at the most boring function known to science:
f : R → R give by f (x) = 0 for all x.
Only marginally more exciting are the constant functions, e.g.,
f : R → R give by f (x) = 5 for all x.
(23/30)
1. Linear functions
(24/30)
A linear function is one whose graph is a straight line. Its equation is
given by
l(x) = mx + b,
where m is the slope, and b is the y -intercept.
Examples:
Graphing linear functions is quite easy:
Examples:
2. Polynomials
(25/30)
Slightly more exciting than the linear functions are the polynomials
A function P is called a polynomial if its equation has the form:
P(x) = an x n + an−1 x n−1 + · · · + a2 x 2 + a1 x + a0 ,
where n is a nonnegative integer, and a0 , a1 , . . . , an are real numbers
called the coefficients.
If an 6= 0 then we say that “P has degree n”.
Examples:
2. Polynomials
(26/30)
Other than constants and linear functions, the most important examples
of polynomial functions are the quadratics and cubics:
Examples:
2. Polynomials
(27/30)
Example: Here is the graph of a cubic polynomial. What is its equation?
3
2
y
1
0
−1
−2
−2
−1
0
x
1
2
3
Exercises
(28/30)
Q1. Find the domain and largest possible range for the following
functions, and sketch them:
(i) f (x) = x|x|
(ii) f (x) =
1
|x| + 1
(iii) f (x) =
1
|x + 1|
(
(iv) f (x) =
x +1
if x < 0
1−x
if x ≥ 0.



x + 9
(v) f (x) = −2x


−6
if x < −3
if |x| ≤ 3
if x > 3.
Exercises
(29/30)
Q2. Solve the following inequality:
|2x + 5| + 20 ≥ 25.
Q3. For each of the following functions, determine if it is even, odd, or
neither.
x
x2 + 1
x2
f (x) = 4
x +1
f (x) = x|x|
t 3 + 3t
f (t) = 4
t − 3t 2 + 4
f (x) = 2 + x 2 + x 4
(i) f (x) =
(ii)
(iii)
(iv)
(v)
Q4. Are the trigonometric functions sin, cos and tan even, odd, or
neither?
Exercises
(30/30)
Q5. Here is the graph of a quadratic
polynomial. What is its
equation?
Q6. The following graph is of a
cubic polynomial whose
equation is
y = K (x − a)(x − b)(x − c).
4
3
Find a, b, c and K .
2
y
1
2
0
1
−1
0
−2
y
−1
−3
−4
−3 −2 −1
0
x
1
2
3
−2
−3
−3
−2
−1
0
x
1
2
3

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