Functions
Lecture: Functions
➢ Terminology
➢ Mapping – 4 types of relationship
➢ Composite functions
➢ Inverse Functions
Definition
• Mapping – relationship between 2 sets of numbers
• Domain – the set of ‘inputs’
• Range – in set of ‘outputs’
4 types
1) 1-1 relationship 2) m-1relationships
3) 1-m relationships 4) m-m relationships
Note: Only 1-1 and m-1 relationships are functions
Notes:
1) f(-x) = f(x) for all values of x are called even function.
e.g. f(x) = x2
2) f(-x) = -f(x) for all values of x are called odd function
e.g. f(x) = x3
Example 1
Given f(x) = 3x2+2. find the value of f(3), f(-2)
E.g. Find the range of each of the following functions for x
i ) f ( x) = x 2 + 4
ii) f ( x) = ( x − 1) 2 + 6
iii) f ( x) = −(2 − x) 2 + 5
iv) f ( x) = 2( x + 4) 2 + 3
E.g. Find their ranges for x > 0
i ) f ( x) = 2 x + 7
ii) f ( x) = −5 x
iii) f ( x) = x − 1
2
iv) f ( x) = ( x + 2) − 3
2
E.g. Find the possible domain of
i ) f ( x) = 4− x
ii ) f ( x ) = x ( x − 4)
1
iii ) f ( x ) =
x−2
1
iv ) f ( x ) =
x−2
Composite Functions
• When 2 or more functions are combined, so
that the output from the first functions
becomes the input to the second function
E.g. f(x) = 2x+1 with domain {1,2,3,4,5} and
g ( x) = x 2 with the domain the range of f
The combined function gf
gf ( x) = (2 x + 1) 2
1) f is applied first, then followed by g
2) Composite function gf can be formed only if the range of f
is a subset of the domain of g.
ff(x)
gf(x)
ff(x) gf(x)
Inverse Functions
• f(x) =2x+1 with
domain{1,2,3,4,5},
range {3,5,7,9,11}
The inverse function of f maps from the range of f
back to the domain
➔ Working backward
x −1 x −1
f −1 ( x) = or f −1 : x
2 2
• To find inverse function
1) Let y = f(x)
2) Rearrange to give x in terms of y
3) Rewrite as f-1(x) by replacing y by x
or
1) Let f(x) = y
2) Change x to y and y to x
3) Rearrange to get y in terms of x
4) Write in the correct form.
f(x) = 3x - 4 y= x
f-1(x) = (x+4)/3
Note:
1) 1-1 functions have inverses.
2) For m-1 functions, an inverse can be defined by restricting
the domain ➔ for that part of the domain, the function is 1-1
E.g. f(x)= x2, f(x) = x2 – 4x
3) The graphs of a functions and its inverse are reflections of
each other in y = x
4) Range of f(x) = Domain of f-1(x)
5) Domain of f(x) = Range of f-1(x)
E.g.
The function f : x x − 2 + 3 has domain x and x > 2.
(a)Determine the range of f
(b)Find the inverse function f -1 and state its domain and range.
(c)Sketch the graphs of y = f(x) and y = f -1(x)
May/June 2022 P12
May/June 2022 P11
Translations
The graph of y = f(x) + a is a translation of the graph y = f(x) by
the vector 0
.
a
Eg.
The graph of y = f(x-a) is a translation of the graph y = f(x) by the vector a
.
0
The graph of y = f(x-a) + b is a translation of the graph y = f(x) by the vector a
.
b
Reflections
The graph of y = -f(x) is a reflection of the graph y = f(x) in the x-axis.
The graph of y = f(-x) is a reflection of the graph y = f(x) in the y-axis.
Stretches
The graph of y = af(x) is a stretch of
The graph y = f(x) with stretch
factor a parallel of the y-axis.
Combined transformations
The graph of y = af(x) + k.
Vertical transformations follow the ‘normal’ order of operations, as used in arithmetic.
The graph of y = f(bx+c).
Horizontal transformations follow the ‘opposite’ order of operations, as used in arithmetic.
Find two different ways of describing the sequence of
transformations the maps the graph of y = f(x) onto the graph
of y = f(2x+10)
June 2020 P13
(a) y = f(-x) (b) y = 2f(x) (c) y = f(x+4) - 3
May/June 22 P13
May/June 22 P11
June 2020 P11 (a) -1 ≤ f(x) ≤ 2
(b) k = 1; translation by 1 unit upwards // to y-axis
(c) y = -3/2 cos 2x -1/2
June 2020 P12 (a) -1 ≤ f(x) ≤ 5; -10 ≤ g(x) ≤ 2
(c) Reflect in x-axis.
Stretch by factor 2 in the y-direction
Translation by –π in the x-direction
