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Functions
& relations
Clement Radcliffe,Contributor
We
completed last week the review of
algebra. Much time was spent on this
and I do recommend mastery in all
areas. Again, I am urging you to proceed
to study with systematic and ongoing
practice. Let us now continue with
the review of aspects of functions
and relations.
Points
to note
(With
respect to the Cartesian Diagram)
- Domain
refers to x values
- Range
refers to y values.
- Function
is a relation in which each element
in the domain (x values) is mapped
on to one and only one element in
the range (y values).
Function
is usually denoted by the symbol f
or g. If y is a function of x, then
the function of x is denoted as f(x)
or g(x). If y is defined such that
y = 2x - 7, then this is represented
as follows:
y
= f(x) = 2x - 7 or f : x 2x - 7
The
latter means The function f such that
x is mapped on to 2x - 7.
The
function is represented on the Cartesian
Diagram by a plot of the equation
y = 2x - 7. All rules related to graphs
and which were indicated previously
must be observed.
Image
of x
This
is the value of f(x) for a given value
of x.
It is found by either reading the
value off the graph or by substituting
into the equation.
Example
Given
that f(x) = 5x - 3, calculate f(-2).
[f(-2) is the value of
f(x)
for which x = -2].
Since
f(x) = 5x - 3
f(-2)
= 5 x -2 - 3 = - 10 - 3 = -13.
Note
that -2 is substituted for x in f(x).
Now
please try the following:
The
function g is defined by g: x x2,
find g(-4).
If
your answer is 16, then you are correct.
Composite
Function
Given
the functions f(x) and g(x), then
the composite function f g(x) is the
function obtained by the function
g(x) being initially applied, followed
by function f (x). In evaluating the
composite function we determine the
function g(x) which is then substituted
for x in f(x).
Points
to note
- It
is important to note that for f
g(x), g(x) replaces x in f(x), while
for g f(x), then f(x) replaces x
in g(x). Note the order well.
- A
common error made by some students
is to find the product of f(x) and
g(x). Avoid this, please.
This
topic is fairly routine and so all
students are encouraged to take full
advantage of the marks allotted to
this problem. In this regard, please
attempt the following:
Example
Given
that f(x) = 1/2x and g(x) = x - 2,
calculate:
(i)
g(-2)
(ii)
f(-7)
(iii)
fg(x)
(iv)
gf(4).
Solution
(i)
Given that g(x) = x - 2, then g(-2)
= -2 - 2 = - 4.
g(-2)
= - 4.
(ii)
Given that f(x) = ?x, then f(-7) =
-7⁄2
f(-7)
= -7⁄2
(iii)
From the definition of f(x) and g(x):
fg(x)
= f(x - 2)
Here
g(x) = x - 2 replaces x in f(x).
f(x
- 2) = ( x-2 )/2
fg(x)
= ( x - 2 )/2
(iv)
As f(x) = x⁄2
................ f(4) = 4⁄2
= 2.
gf(4)
= g(2)
As
g(x) = x - 2, g(2) = 2 - 2 = 0.
gf(4)
= 0.
Alternatively
Given
the definition of f and g:
gf(x)
= g x⁄2
As
g(x) = x - 2 ................. g x⁄2
= x⁄2
- 2
Simplifying,
x⁄2 -
2 = ( x - 4 )/2
gf(x)
= ( x - 4 )/2
gf(4)
= ( 4 - 4 )/2 = 0.
Let
us attempt another example:
Given
that f(x) = x + 2 and g(x) = 3⁄x
,
(i)
calculate f(-1)
(ii)
write an expression for gf(x)
(iii)
calculate the values of x so that
f(x) = g(x).
Solution
(i)
Since f(x) = x + 2 f(-1) = -1 + 2
= 1.
f(-1)
= 1
(ii)
Given the values of f(x) and g(x)
gf(x)
= g(x + 2)
gf(x)
= 3/x
+ 2
Note
In
the composite function gf(x), f(x)
replaces x in g(x).
(iii)
Given that f(x) = g(x)
x
+ 2 = 3⁄x
Simplifying
by multiplying both sides by x.
x
x( x + 2) = x x 3⁄x
x2
+ 2x =3
x2
+ 2x -3 = 0
Solve
the quadratic equation using the factorisation
method:
(x
+ 3)(x - 1) = 0
x
+ 3 = 0
x
= - 3.
OR
x
- 1 = 0
x
= 1.
Answer:
x = - 3 or x = 1.
Homework
Given
that f : x -------------> 3x -
2
g : x ----------------------->
2x + 5
Evaluate:
(i)
g(-6)
(ii)
fg (3)
If
f(x) = 2x - 1 and g(x) = ? (x + 2),
calculate
(i)
f(3)
(ii)
gf(3).
Enjoy
your week.
Clement
Radcliffe is an independent contributor.
Send questions and comments to kerry-ann.hepburn@gleanerjm.com
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