Functions and relations
Clement Radcliffe, Contributor

Students of Wolmer's Girls' School in Kingston, line both sides of Marescaux Road, in front of the school, on Wednesday, February 1, to pay their last respect as the hearse with the body of their former senior vice-principal, Peggy Harrison, drives by. - Norman Grindley Photo

WE COMPLETED last week, the review of algebra. Much time was spent on this and I do recommend mastery in all areas. Please study with systematic 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 onto 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 = 2 x - 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 onto 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) = 4x - 3, calculate f(-2). [f(-2) is the value of f(x) for which x = -2].

Since f(x) = 4x - 3

... f(-2) = 4 x -2 - 3 = - 8 - 3 = -11.

Note that -2 is substituted for x in f(x).

Now please try the following:

The function g is defined by g: x 2x², find g(-4).

If your answer is -32, then you are correct.

COMPOSITE FUNCTION

Given the functions f(x) and g(x), then the composite function fg(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 the f(x), while for g f(x), then f(x) replaces x in g(x). Note the order as 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(3)
(ii) f(-7)
(iii) fg(x)
(iv) gf(4).

Solution

(i) Given that g(x) = x - 2, then g(3) = 3 - 2 = 1.
... g(3) = 1.

(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)

(iv) As f(x) = x/2 ... f(4) = 4/2 = 2.

As g(x) = x - 2, ...g(2) = 0.
.. gf(4) = 0.

Alternatively

Given the definition of f and g:

... gf(x) = g x/2 = x/2 - 2

Now please attempt the following for homework:

* Given that f(x) = 3x and g(x) = x - 2, calculate gf(2).

* A composite function K is defined as K(x) = (2x - 1)².

Express K(x) as gf(x), where f(x) and g(x) are two simple functions.

* The function f and g are defined as follows:

f(x) = 2x² - 5

g(x) = 3x - 2

Evaluate

a) f(-3) b) gf(-3)

Enjoy your week.

* Clement Radcliffe is principal of Glenmuir High School in Clarendon.