Lecture Notes Basic Functions and their Properties page 1
De…nition: A function f is one-to-one (or injective) if for all a and b in its domain, if a 6= b, then
f (a) 6= f (b).
Alternative de…nition: A function f is one-to-one (or injective) if for all a and b in its domain, if
f (a) = f (b), then a = b.
De…nition: A function f is increasing on an interval I if for all a and b in I, if a < b, then f (a) f (b).
De…nition: A function f is strictly increasing on an interval I if for all a and b in I, if a < b, then
f (a) < f (b).
De…nition: A function f is decreasing on an interval I if for all a and b in I, if a < b, then f (a) f (b).
De…nition: A function f is strictly decreasing on an interval I if for all a and b in I, if a < b, then
f (a) > f (b).
De…nition: A function f is even if for all x in its domain , f ( x) = f (x). The graph of an even function
is symmetrical to the y axis.
De…nition: A function f is odd if for all x in its domain , f ( x) = f (x). The graph of an odd function
is symmetrical to the origin.
Note that while an integer is either even or odd, most functions are neither even, nor odd. Even and odd
functions are sort of rare but these properties are very useful for us.
De…nition: A rational function is a quotient of two polynomial functions.
The concept of a continuous function is very important. Although this term will not be precisely
de…ned, the intuitive idea of a continuous function is thatwe can draw its graph without lifting the pencil.
For example, f (x) = x2
is a continuous function but g (x) =
1
x
is not; it is not continuous at x = 0.
c Hidegkuti, Powell, 2011 Last revised: October 29, 2016
Lecture Notes Basic Functions and their Properties page 2
Basic Functions
1.) Linear functions f (x) = mx + b where m 6= 0.
The graph is a straight line. One useful form of the equation can be obtained by factoring out the slope:
f (x) = mx + b = m x +
b
m
x
y
x
y
m > 0
x
y
x
y
m < 0
Case 1. If m > 0 Case 2. If m < 0
domain: R domain: R
range: R range: R
y intercept: (0; b) y intercept: (0; b)
x intercept:
b
m
; 0 x intercept:
b
m
; 0
one-to-one one-to-one
no maximum or minimum no maximum or minimum
strictly increasing strictly decreasing
continuous on R continuous on R
c Hidegkuti, Powell, 2011 Last revised: October 29, 2016
Lecture Notes Basic Functions and their Properties page 3
2.) Quadratic functions f (x) = ax2
+ bx + c where a 6= 0.
The graph is a parabola. It opens upward if a > 0 and opens downward if a < 0.
x
y
x
y
a > 0
x
y
x
y
a < 0
Case 1. If a > 0 Case 2. If a < 0
Example: f (x) = x2
+ 4x 5 Example: f (x) =
1
2
x2
+ 3x
5
2
standard form: f (x) = (x + 2)2
9 standard form: f (x) =
1
2
(x 3)2
+ 2
factored form: f (x) = (x + 5) (x 1) factored form: f (x) =
1
2
(x 1) (x 5)
domain: R range: [ 9; 1) domain: R range: ( 1; 2]
y intercept: (0; 5) y intercept: 0;
5
2
x intercepts: ( 5; 0) and (1; 0) x intercepts: (1; 0) and (5; 0)
not one-to-one not one-to-one
no maximum no minimum
minimum: ( 2; 9) maximum: (3; 2)
strictly decreasing on ( 1; 2) strictly increasing on ( 1; 3)
strictly increasing on ( 2; 1) strictly decreasing on (3; 1)
continuous on R continuous on R
c Hidegkuti, Powell, 2011 Last revised: October 29, 2016
Lecture Notes Basic Functions and their Properties page 4
3.) Monomials f (x) = xn
43210-1-2-3-4
7
6
5
4
3
2
1
0
-1
x
y
x
y
n is even
43210-1-2-3-4
4
3
2
1
0
-1
-2
-3
-4
x
y
x
y
n is odd
Case 1. If n is even Case 2. If n is odd
domain: R range: [0; 1) domain: R range: R
y intercept: (0; 0) y intercept: (0; 0)
x intercept: (0; 0) x intercept: (0; 0)
not one-to-one one-to-one
no maximum no minimum or maximum
minimum: (0; 0) strictly increasing on R
strictly decreasing on ( 1; 0) continuous on R
strictly increasing on (0; 1)
continuous on R black graph: f (x) = x3
red graph: f (x) = x5
black graph: f (x) = x2
red graph: f (x) = x4
c Hidegkuti, Powell, 2011 Last revised: October 29, 2016
Lecture Notes Basic Functions and their Properties page 5
4.) The rational functions f (x) =
1
x
and g (x) =
1
x2
543210-1-2-3-4-5
5
4
3
2
1
0
-1
-2
-3
-4
-5
x
y
x
y
f (x) =
1
x
543210-1-2-3-4-5
8
7
6
5
4
3
2
1
0
-1
-2
x
y
x
y
g (x) =
1
x2
domain: R n f0g range: R n f0g domain: R n f0g range: (0; 1)
no y intercept no y intercept
no x intercept no x intercept
one-to-one not one-to-one
no maximum or minimum no minimum or maximum
strictly decreasing on ( 1; 0) and on (0; 1) strictly increasing on ( 1; 0) and
strictly decreasing on (0; 1)
not continuous at x = 0 not continuous at x = 0
vertical asymptote: the line x = 0 vertical asymptote: the line x = 0
horizontal asymptote: the line y = 0 horizontal asymptote: the line y = 0
c Hidegkuti, Powell, 2011 Last revised: October 29, 2016
Lecture Notes Basic Functions and their Properties page 6
5.) Radical functions f (x) = n
p
x
9876543210-1
9
8
7
6
5
4
3
2
1
0
-1 x
y
x
y
n is even
43210-1-2-3-4
4
3
2
1
0
-1
-2
-3
-4
x
y
x
y
n is odd
Case 1. If n is even Case 2. If n is odd
domain: [0; 1) range: [0; 1) domain: R range: R
y intercept: (0; 0) y intercept: (0; 0)
x intercept: (0; 0) x intercept: (0; 0)
one-to-one one-to-one
no maximum no minimum or maximum
minimum: (0; 0) strictly increasing
strictly increasing continuous on R
continuous on (0; 1)
black graph: f (x) = 3
p
x
black graph: f (x) =
p
x red graph: f (x) = 5
p
x
red graph: f (x) = 4
p
x
c Hidegkuti, Powell, 2011 Last revised: October 29, 2016
Lecture Notes Basic Functions and their Properties page 7
6.) Exponential functions f (x) = ax
where a > 0.
543210-1-2-3-4-5
9
8
7
6
5
4
3
2
1
0
-1 x
y
x
y
a > 1
543210-1-2-3-4-5
9
8
7
6
5
4
3
2
1
0
-1 x
y
x
y
0 < a < 1
Case 1. If a > 1 Case 2. If 0 < a < 1
domain: R range: (0; 1) domain: R range: (0; 1)
no x intercepts no x intercepts
y intercept: (0; 1) y intercept: (0; 1)
one-to-one one-to-one
no maximum or minimum no maximum or minimum
strictly increasing strictly decreasing
horizontal asymptote: y = 0 horizontal asymptote: y = 0
continuous on R continuous on R
c Hidegkuti, Powell, 2011 Last revised: October 29, 2016
Lecture Notes Basic Functions and their Properties page 8
7.) Logarithmic functions f (x) = loga x where a > 0 and a 6= 1.
9876543210-1
5
4
3
2
1
0
-1
-2
-3
-4
-5
x
y
x
y
a > 1
9876543210-1
5
4
3
2
1
0
-1
-2
-3
-4
-5
x
y
x
y
0 < a < 1
Case 1. If a > 1 Case 2. If 0 < a < 1
domain: (0; 1) range: R domain: (0; 1) range: R
no y intercepts no y intercepts
x intercept: (1; 0) x intercept: (1; 0)
one-to-one one-to-one
no maximum or minimum no maximum or minimum
strictly increasing strictly decreasing
vertical asymptote: x = 0 vertical asymptote: x = 0
continuous on (0; 1) continuous on (0; 1)
8.) Absolute value function, f (x) = jxj
43210-1-2-3-4
5
4
3
2
1
0
-1
x
y
x
y
domain: R range: [0; 1)
x intercept: (0; 0) ; y intercept: (0; 0)
not one-to-one
no maximum
minimum: (0; 0)
strictly decreasing on (1; 0) and strictly increasing on (0; 1)
continuous on R
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c Hidegkuti, Powell, 2011 Last revised: October 29, 2016