Functions and relations
Clement Radcliffe,Contributor

Last week we completed 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 is now continue with the review of aspects of functions and relations.
The following is done with respect to the Cartesian Diagram. It is deliberate that this review comes after graphs.

Points to note

• 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 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 finctions 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 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). Since f (x) = x/2
... 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). 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) As g(x) = 3/x
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 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.

As 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.

Clement Radcliffe is principal of Glenmuir High School. Send questions and comments to kerry-ann.hepburn@gleanerjm.com