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Solutions
Clement Radcliffe, Contributor
 |
| Cornwall
College students, Sanjay Thorpe and Renaldo Harvey, try their hands at the turntables
during FAME FM's FAME School Rules, recently. They are keenly observed by FAME
disc jockey, Marlon Young. - Photo by Tashieka Mair | We
began the review of functions and relations, last week. In today's lesson, we
will share the solution to the homework given. Given
that f(x) = 3x and g(x) = x - 2, calculate gf(2). Solution:
As f(x)
= 3x and g(x ) = x - 2
ie.
gf (x) = g(3x) f(x) replaces x in g(x)
ie.
g(3 x ) = 3x - 2.
ie.
gf(2) = 3 x 2 - 2 = 4.
ie.
gf(2) = 4. 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 ie.
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 | ie. |
ax = y - b | | ie. | x
= | y
- b | (x
is expressed as a function of y) | | | | a | | | Interchange
x for y. | (This
is necessary, as y is always expressed as a function of x ) | | | | | | ie. | x
- b | | | | a | | | | | | | | | ie.
f-1
(x) | x
- b | ie.
f-1 = x
- b | | | | a | | | a | | | | | | | | | | that
is, the inverse of function f is x
- b | | | | | | | a | | | | | | | | | | Please
note that this method should always end with the statement: | | f-1
(x) = x
- b | and
NEVER y = x
- b | | | | a | | | a | | |
| Given
the function y = ax + b, some students express f-1
(x) as |
1 | | | a
x + 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)
Since
f(x) = 1/2(x + 2)
ie.
y = 1/2(x + 2) 2y
= x + 2
ie.
x = 2y - 2 Interchanging
x for y, (Always remember this step; it must also be explicitly stated.) ie.
y = 2x - 2
ie.
f-1 ( x ) =
2 x - 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) = 3 x and g(x) = x - 2, determine the functions: (a)
fg(x) (b) fg -1 (x) Solution:
a)
As f(x) = 3x and g(x) = x - 2 ie.
fg(x) = f(x - 2) = 3(x - 2) ie.
fg(x) = 3(x - 2) b)
y = fg(x) = 3(x - 2) ie.
y = 3x - 6 ie.
3x = y + 6 | ie.
x = | x
+ 6 | Interchange
x for y | | | 3
| | | | | | ie. | x
+ 6 | | | | 3
| | | | | | | ie.
The inverse of fg(x) OR (fg)-1
(x) is | x
+ 6 | | | 3 |
Let
us attempt another example: Example
Given
f(x) = x²
and g(x) = 5x + 3, calculate (i)
f (-2) (ii)
gf (-2) (iii)
(gf)-1 x Solution
(i)
Since f(x) = x²
f ie.(-2) = (-2) 2 = 4. Answer:
f (-2) = 4. (ii)
gf(-2) = 4. As f(-2) = 4 Since
g(x) = 5x + 3 gf(-2) = g(4) = 5 x 4 + 3 = 23 ie.
gf(-2)= 23 (iii)
Given the values of f(x) and g(x) then
gf(x) = g(x²)
Since
g(x) = 5x + 3 g(x²)
= 5x²
+ 3 ie.
gf(x) = 5x²
+ 3 In
order to find the inverse, then let y = gf(x) ie.
y = 5x²
+ 3 ie.
5x² =
y - 3 
Please
do the following for homework. Prove
that if g: x  2x- 1
then g-1 is | x
+ 1 | | | 2 |
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 (gf)-1
(x). Have
a good week. Clement
Radcliffe is the principal of Glenmuir High School in May Pen. | 
