Functions and relations - solution
Clement Radcliffe, Contributor

Norma Rochester, coordinator for the National Schools' Debate Competition, hands over a trophy to students of the Green Island High School in Westmoreland. From left are Natoya Bramwell, Nerrick Dias and Keline Maye, head girl. - File

WE BEGAN the review of functions and relations last week. In today's lesson, we will share the solution to last week's homework.

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

SOLUTION:

As f(x) = 3x and g(x) = x - 2
... gf(x) = g(3x) f(x) replaces x in g(x)
... g(3x) = 3x - 2.
... gf(2) = 3x 2 - 2 = 4.
... gf(2) = 4.

2. The function f and g are defined as follows:

f(x) = 2x² - 5
g(x) = 3x - 2

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

Solution:

(a) Since f(x) = 2x² - 5

... f(- 3) = 2(- 3)² - 5 = 2 x 9 - 5 = 13
... f(- 3) = 13

(b) As g(x) = 3x - 2 and f(- 3) = 13

Then gf(- 3) = g(13) f(x) replaces x in g(x)
... g(13) = 3 x 13 - 2 = 37
... gf (- 3) = 37

3. 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.

Solution:

If K(x) = gf(x) = (2x - 1)².

By inspection, if t = 2x - 1, then K(x) = t2

... f(x) = 2x - 1 and g(x) = x2.

Now that we have gone through the homework, our lesson will continue.

INVERSE OF A FUNCTION

If f is the function defined as

y = ax + b

Then f -1, the inverse function, expresses the variable x in terms of y.

Example: y = ax + b

... ax = y - b

Interchange x for y. (This is necessary, as y is always expressed as a function of x)

that is, the inverse of function f is

Please note that this method should always end with the statement:

Given the function of y = ax + b, some students express f -1(x) as 1/ax + b by assuming that -1 is the power of f as in indices. I am sure you will never make this error.

Example: Given that f(x) = 1/2(x + 2). Calculate f -1(x)

Calculate f-1(x)
Since f(x) = 1/2(x + 2)
... y = 1/2(x + 2)
2y = x + 2
... x = 2y - 2

Interchanging x for x, (Always remember this step; it must also be explicitly stated.)

... y = 2x - 2
... f -1(x) = 2x - 2

Please be sure that you are comfortable with the methods of cross-multiplication and changing the subject of a formula.

INVERSE OF A COMPOSITE FUNCTION

Given the functions y = f(x) and y = g(x), then y = gf(x) is a composite function.

Since gf(x) is a function of x, the inverse is found by using the method outlined above.

Example: Given the functions f(x) = 3x and g(x) = x - 2, determine the functions:

(a) fg(x)
(b) [fg(x)]-1

Solution:

(a) As f(x) = 3x and g(x) = x - 2
... fg(x) = f(x - 2) = 3(x - 2)
... fg(x) = 3(x - 2)

(b) y = fg(x) = 3(x - 2)
... y = 3x - 6
... 3x = y + 6

Interchange x for y

The inverse of fg(x) is

Please do the following for homework.

* Prove that if g: x 2x - 1 then g-1 is

* If f and g are defined as follows:

f:x 3x - 5 and g:x 1/2 x

(a) Calculate the value of f (3)

(b) Write expressions for

(i) f -1 (x) (ii) g -1 (x)

(c) Hence, or otherwise, write an expression for (g f)-1

Enjoy your week.

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