FUNCTION NOTATION
f:x |----> 2x can be written as f(x) = 2x
Example 1
Given function f:x |---> 3x-2, find the
(a) Image of -1
(b) object which has the image 4
Answer
(a) Image of -1, which is mean that -1 is a object. So substitute "x" as a -l.
f:x |---> 3x-2
f(x) = 3x-2
f(-1) = 3(-1)-2 substitute x = -1
f(-1) = -5, so the image of -1 is -5
(b) object which has the image 4. (number 4 is a image)
3x-2 = 4
3x = 4+2
3x = 6
x = 6/3
x = 2, so the object of 4 is 2
COMPOSITE FUNCTIONS
Example 1
The functions of f and g are defined as f:x |----> 3x +1 and
g:x |----> x/2
find
(a) the composite functions of gf and the value of gf(4),
(b) the composite function fg and the value of fg(4).
Answer
(a) Given f:x |----> 3x +1 and g:x |----> x/2
so f(x) = x +1 and g(x) = x/2
Solution 1
gf(x) = g(f(x))
First, substitute f(x) = 3x +1
gf(x) = g(3x+1), since g(x) = x/2, so the "x" of the g(x) will be 3x+1.
gf(x) = g(3x+1)
gf(x) = (3x+1)/2
gf(4) = [(3(4) + 1)/2]
gf(4) = 13/2#
Solution 2
gf(x) = g(f(x))
gf(x) = g(3x+1)
Given g(x) = x/2
so substitute x of the g(x) = 3x +1
g(x) = x/2
gf(x) = (3x+1)/2
gf(4) = (3[4]+1)/2
gf(4) = 13/2#
(b) f(x) = 3x +1 and g(x) = x/2
fg(x) = ?
Solution 1
fg(x) = f(g(x)), first, substitute g(x) = x/2
fg(x) = f(x/2)
Given f(x) = 3x +1, replace x = x/2
so fg(x) = 3(x/2) + 1
fg(4) = 3(4/2) + 1 = 7
Solution 2
fg(x) = f(g(x)) , g(x) = x/2
fg(x) = f(x/2)
Given f(x) = 3x +1
So: fg(x) = 3(x/2) + 1
fg(4) = 3(4/2) + 1 = 7#
INVERSE FUNCTION
Example 2
The arrow diagram represents a function f:x ---- > px + q/x, x ≠
0.
Find
(a) the values of p and of q.
(b) the image of 2 under the function
Answer
(a) the values of p and of q.
** Substitute
the image and object into the equation
** X is the
object, f(x) image
Object Image
1 -2
4 7
f(x) = px + q/x
Equation 1 Equation
2
-2 = p(1) + q/1 7
= p(4) + q/4
p + q = -2 4p
+ q/4 = 7
16p + q = 28
p + q = -2 ------------ 1
16p + q = 28 ------------2
q-q = 0
So equation 1- equation 2
p-16p = -2 – 28 So,
p + q = -2
-15p = -30 2
+ q = -2
p = 2 q = -4
(b) the image of 2 under the function
f(x) = px + q/x
p = 2, q =
-4
f(x) = 2x
-4/x
** Image of 2? Means 2 is the object and the object is the x
f(x) = 2x -4/x
2(2) – 4/2 = 2
So the image is 2.
Example 3:
The arrow diagram represents a function
f:x |-----> x2
+bx + c, find the values of b and c.
Answer
f(x) = x2 +bx + c
** Substitute the image and object into the equation
** X is the object, f(x) image
Object Image
-1 -6
2 6
f(x) = x2 +bx + c
Equation 1 Equation 2
-6 = (-1)2 +b(-1) + c 6 = (2)2 +b(2) + c
1 – b + c = -6 4
+ 2b + c = 6
-b + c = -7 2b + c = 2
b – c = 7
b – c = 7 -------------1
2b + c = 2 -------------2
-c + c = 0
So: Eq 1 + Eq 2
b + 2b = 7 + 2 When
b = 3
3b = 9 So:
b – c = 7
b = 9/3 3 – c = 7
b = 3 c
= -4
COMPOSITE FUNCTIONS
Example 1
The functions of f and g are defined as f:x |----> 3x +1 and
g:x |----> x/2
find
(a) the composite functions of gf and the value of gf(4),
(b) the composite function fg and the value of fg(4).
Answer
(a) Given f:x |----> 3x +1 and g:x |----> x/2
so f(x) = x +1 and g(x) = x/2
Solution 1
gf(x) = g(f(x))
First, substitute f(x) = 3x +1
gf(x) = g(3x+1), since g(x) = x/2, so the "x" of the g(x) will be 3x+1.
gf(x) = g(3x+1)
gf(x) = (3x+1)/2
gf(4) = [(3(4) + 1)/2]
gf(4) = 13/2#
Solution 2
gf(x) = g(f(x))
gf(x) = g(3x+1)
Given g(x) = x/2
so substitute x of the g(x) = 3x +1
g(x) = x/2
gf(x) = (3x+1)/2
gf(4) = (3[4]+1)/2
gf(4) = 13/2#
(b) f(x) = 3x +1 and g(x) = x/2
fg(x) = ?
Solution 1
fg(x) = f(g(x)), first, substitute g(x) = x/2
fg(x) = f(x/2)
Given f(x) = 3x +1, replace x = x/2
so fg(x) = 3(x/2) + 1
fg(4) = 3(4/2) + 1 = 7
Solution 2
fg(x) = f(g(x)) , g(x) = x/2
fg(x) = f(x/2)
Given f(x) = 3x +1
So: fg(x) = 3(x/2) + 1
fg(4) = 3(4/2) + 1 = 7#
INVERSE FUNCTION
Example 1
The function f is defined as f:x |---- > 2x-5. Find
(a) f-1(3)
(b) f-1(x).
Answer
Given f(x) = 2x-5
*First step: Let find inverse function of the f which is f-1(x),
the substitute x=3
How to find inverse function? It is just let y = 2x-5,
then let x be alone and change the “y” to “x”.
y = 2x-5
2x = y +5
X = (y+5)/2
f-1(x) = (x+5)/2#
So the inverse of f is (x+5)/2. So the question (b) is answered. So
for the question (a), just substitute x =3
f-1(x) = (x+5)/2
f-1(3) = (3+5)/2
f-1(3) = 4#
Example 2
f:x |---- > (x-3)/(2x+1) and g: x |---- > 4x. Find
(a) f-1(x) (b)
f-1g(x) (c) gf-1(x)
Answer
(a) f-1(x)
|
(b) f-1g(x)
|
(c) gf-1(x)
|
Given f(x)
= (x-3)/(2x+1)
Y = (x-3)/(2x+1)
Y(2x+1) =
x-3
2xy + y =
x-3
2xy – x =
-3 –y
x(2y-1) =
-3-y
x =
(-3-y)/(2y-1)
f-1(x)
= (-3-x)/(2x-1)
or
f-1(x)
= -(3+x)/(2x-1)
f-1 (x) = (3 + x)/ (-2x + 1)
f-1 (x)
= (x+3)/(1-2x)#
|
f-1 (x)
= (x+3)/(1-2x)
Given g(x)
= 4x
f-1g(x)
so f-1(4x),
*Substitute x of f-1 = 4x
f-1(4x)
[4x+3]/ [1-2(4x)]
= [4x+3]/[1-8x]
So: f-1g(x) = [4x+3]/[1-8x]#
|
f-1 (x) =
(x+3)/(1-2x)
gf-1(x)
g[(x+3)/(1-2x)]
*Substitute
x of g = (x+3)/(1-2x)
g[(x+3)/(1-2x)]
4[(x+3)/(1-2x)]
= (4x+12)/(1-2x)
So: gf-1(x) = (4x+12)/(1-2x)
|
Example 3
Given f(x) = (x+3)/3 and f-1 (x) = px + q. Find
(a) the values of p and q,
(b) the value of
f-1(-3)
Answer
(a) the
values of p and q
|
(b) the
value of f-1(-3)
|
f(x) =
(x+3)/3
y =
(x+3)/3
3y = x+3
x = 3y -3
f-1(x)
= 3x-3
Given f-1
(x) = px + q
*Compare
the both equation
3x-3 = px
+ q
3x = px -3 = q
p = 3 q
= -3
|
f-1(x)
= 3x-3
f-1(-3)
So: f-1(-3)
= 3(-3) – 3
f-1(-3) = -12#
|
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