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Functions
and 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
of the graph or by substituting it
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) = 1/2x, 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
(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 ang 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) CXC, January 2001,
5(b)
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
NB.
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 + 2) = x * 3/x
...
x2 + 2x = 3
... x2 + 2x - 3 = 0
Solve
the quadratic equation using the factorization
method:
...
(x + 3)(x - 1) = 0
...
x + 3 = 0
...
x = -3
OR
x - 1 = 0
...
x = 1
Answer:
x = -3 or x = 1
A
usual, I close with your 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) = 1/2(x + 2)
Calculate
(i)
f(3)
(ii)
gf(3)
Enjoy
your week.
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John
Martin, a former drug user and
deportee, talks to students
of class 8H at the Mona High
School in St Andrew about substance
abuse, rape and other issues
that teens are affected by in
this 2006 photograph.
- Ricardo Makyn/Staff Photographer
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Clement
Radcliffe is the principal of Glenmuir
High School in May Pen.
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