Inverse Functions and Logarithmic Functions - Desklib

INVERSE FUNCTIONS
Inverse of the graph ( y=10x ¿
If a function f ( x ) ismapping x ¿ y ,then theinverse function of f ( x ) maps y back ¿ x
f ( x ) =10x ,interchan gingthe variables x∧ y , we have :
x=10 y
Thus we solve for y .
Given that f ( x ) =g ( x ) ,then ln f ( x ) =ln g ( x )
ln x=ln 10 y
Since log a xb =b∗log a x
ln 10 y= y∗ln10
ln x= y∗ln 10
y= ln x
ln10
Thus the inverse of y=10x is y = ln x
ln10 .
Hence we graph y= ln x
ln 10
To graph the inverse function, we determine (x,y) points.

f−1 ( x ) if f ( x ) =−4 x +9
If a function f ( x ) ismapping x ¿ y ,then theinverse function of f ( x ) maps y back ¿ x
y=−4 x +9
x=−4 y +9
−4 y +9= x
−4 y +9−9=x−9
−4 y=x−9
−4 y
−4 = x
−4 =−9
−4
y=−x−9
4
f−1 ( x )=−x−9
4
Inverse of f ( x )= ( x +1 )2−3
Simplifying f ( x ) = ( x +1 ) 2−3
f ( x ) =x2 +2 x−2
If a function f ( x ) ismapping x ¿ y ,then theinverse function of f ( x ) maps y back ¿ x
x= y2 +2 y−2
y2 +2 y−2=x Subtracting x ¿ bothsides , we have :
y2 +2 y−2−x =x−x
y2 +2 y−2=0
Using quadratic formulawith a=1, b=2 , c=−2−x
y= √x +3−1
y=− √x +3−1
Thus Inverse of f ( x ) = ( x+1 ) 2 −3is : y= √ x +3−1∧ y=− √ x +3−1
Inverse of f ( x )=ln 2 x

If a function f ( x ) ismapping x ¿ y ,then theinverse function of f ( x ) maps y back ¿ x
y=ln 2 x
x=ln2 y applying logarithm rule :a=log b ¿
x=ln ex
ln (ex ¿)=ln ( 2 y ) Given the logs have the same base :log b f ( x ) =log g b ( x ) ¿
Implying that f ( x )=g ( x )
For ln ex=ln 2 y ,
we solve ex=2 y
y= ex
2
The inverse of Inverse of f ( x ) =ln 2 x is f ( x ) = ex
2
Inverse of the funct ion 4.5 ( x−15 ) 4 +3 when x ≤15
Applying binomial theorem, we have :
( a+ b )n=∑
i =0
n
(n
i )an−i bi Where a=x ,b=−15
∑
i=0
n
( n
i ) an−i bi=∑
i
4
( 4
i ) x4−i ( −15 ) i
4.5 ( x−15 )4 +3=4.5 ∑
i
4
(4
i ) x4−i (−15 )i +3
f ( x ) =4.5 x4− 67.5∗24
6 x3 + 1012.5∗24
4 x2− 15187.5∗24
6 x +227815.5
If a function f ( x ) ismapping x ¿ y ,then theinverse function of f ( x ) maps y back ¿ x
x=4.5 y4− 67.5∗24
6 y3+ 1012.5∗24
4 y2−15187.5∗24
6 y +227815.5
solving for y , we have 45 y4 −2700 y3 +60750 y2−607500 y +2278155− x∗10=0