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Inverse
of a function
Clement Radcliffe,Contributor
We
began the review of functions and
relations last week. In this week's
lesson we will share the solution
to last week's homework.
- Given
that f : x -----> 3x -
2
g
: x ---------------> 2x + 5
Evaluate:
(i)
g(-6)
(ii)
fg (3)
Solution:
(i)
Since g : x -----> 2x +
5, g(x) = 2x + 5
...
g(-6) = (2 x -6) + 5
=
-12 + 5 = -7.
...
g(-6) = -7
As
g(x) = 2x + 5 and f(x)
= 3x - 2
fg(x)
= f (2x + 5)
fg(x)
= 3 (2x + 5) - 2
=
6x + 15 - 2 = 6x + 13
...
fg(3) = (6 x 3) + 13 = 18 +13
fg(3)
= 31
- If
f(x) = 2x - 1 and g(x) =
1⁄2
(x + 2),
calculate
(i)
f(3)
(ii)
gf(3).
Solution:
(i)
As f(x) = 2x - 1, then f(3)
= (3 x 2) -1 = 6 - 1 = 5.
...
f(3) = 5
(ii)
Since f(3) = 5, then gf(3)
= g(5) .
Since
g(x) = 1⁄2
(x + 2)
g(5)
= 1⁄2
(5 + 2) = 7⁄2
...
gf(3) = 7⁄2
.
Alternatively
f(x)
= 2x - 1 and g(x) = 1⁄2
(x + 2),
gf(x)
= g (2x - 1 )
=
1⁄2 (2x
- 1) + 2)
=
1⁄2 (2x
- 1 + 2)
gf(x)
= 1⁄2
(2x + 1)
gf(3)
= 1⁄2
(2 x 3 + 1)
=
7⁄2
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
...
x = ( y - b )/a ( x is expressed as
a function of y)
Interchange
x for y. (This is necessary as y is
always expressed as a function of
x)
...
y = ( x - b )/a
f-1(x)
= ( x - b )/a or f-1 =
( x - b )/a
that
is, the inverse of function f,
(f-1), is ( x -
b )/a
Please
note that this method should always
end with the statement:
f-1
(x) = ( x - b )/a and NEVER
y = ( x - b )/a.
Given
the function 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
1: Given that f(x)
= 1⁄2
(x + 2). Calculate f-1(x)
Solution
Since f(x) = 1⁄2
(x + 2)
...
y = 1⁄2
(x + 2)
2y
= x + 2
...
x = 2y - 2
Interchanging
x for y, (Always remember this
step; it must also be explicitly stated.)
...
y = 2x - 2
...
f-1(x) = 2x - 2
Example
2: Given f(x) = 1⁄2x
and g(x) = x - 2
Calculate:
(i)
g(-2)
(ii)
fg(4)
(iii)
f-1(4)
Solution
(i) Since g(x) = x - 2, then
g(-2) = -2 -2 = -4
NB
x is replaced by -2 in g(x).
(ii)
Given g(x) = x - 2
then
fg (x) = f(x - 2)
...
fg(4) = f(4 - 2) = f(2)
As
f(x) = 1⁄2x
f(2) = 2⁄2
= 1.
I
am sure that you can now show that
fg(x) = ( x - 2 )/2
(iii)
As f(x) = 1⁄2x
then
y = x⁄2
...
x = 2y. Interchanging x for y
...
y = 2x
...
f-1(x) = 2x
...
f-1(4) = 8
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]-1(x)
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
...
x = ( y + 6 )/3 Interchange x for
y
...
y = ( x + 6 )/3
...
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)
(g f)-1x
Solution
(i)
Since f(x) = x2
... f(-2) = (-2)2
= 4.
Answer:
f(-2) = 4.
(ii)
As seen from above, f(-2) =
4
And
since g(x) = 5x + 3 .... gf(-2)
= g(4) = (5 x 4) + 3 = 23
...
gf(-2)= 23
(iii)
Given that f(x) = x2
and g(x) = 5x + 3 then gf(x)
= g(x2)
Since
g(x) = 5x + 3 g(x2)
= 5x2 + 3
...
gf(x) = 5x2 + 3
(NB.
If gf(x) = 5x2 +
3, then gf(-2) = 5 x (-2)2
+ 3 = 23 as above.)
In
order to find the inverse, then let
y = gf(x)
...
y = 5x2 + 3
...
5x2 = y - 3
...
x2 = ( y - 3 )/5
x
= square root of ( y - 3 )/5
Interchanging
x for y
...
y = square root of ( x - 3 )/5
...
(gf)-1x = square
root of ( x - 3 )/5
Please
do the following for homework.
-
f and g are functions
defined as follows
f
: x -------------> ( x + 1 )/2
g
: x -------------> 2x + 7
(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
- Given
that f : x x + 3 and g : x 2x
(a)
Determine fg-1(x)
and g-1 f-1
(x)
(b)
Hence evaluate fg-1
(5 ) and g-1 f-1
(5)
Have
a good week
Clement
Radcliffe is an independent contributor.
Send questions and comments to kerry-ann.hepburn@gleanerjm.com
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