Under what conditions the function
y = (ax+b)/(cx+d), (ad - bc) = 0 is it’s own inverse?

Question

Under what conditions the function y = (ax+b)/(cx+d), (ad - bc) = 0 is it’s own inverse?

bharat bajaj · Answer

First inverse the function :
x = ay + b / cy + d
xcy + dx = ay + by = b - dx/ cx - a
It should be equal to original function
b - dx/ cx - a = ax + b/cx + d
bcx + bd - dcx^2 - d^2x = acx^2 + bcx - a^2x - ab

By this, we get a = -d should be the other condition.
Thanks
Bharat
askiitians faculty
IIT Delhi

Sher Mohammad · Answer

y = (ax + b) / (cx + d)

To find the inverse, switch x and y, and then solve for y:
x = (ay + b) / (cy + d)
x(cy + d) = ay + b
cxy + dx = ay + b
cxy - ay = b - dx
y(cx - a) = b - dx
y = (b - dx) / (cx - a) # inverse

for both to be equal

(ax+b)/(cx+d)=(b - dx) / (cx - a)

(ax+b)(cx-a)=(b-dx)(cx+d)

acx^2-a^2x+bcx-ab=-dcx^2+bcx-d^2x+bd

(ac+dc)x^2+(d^2-a^2+bc-bd)x-(ab+bd)

imply

c(a+d)=0

a=-d

d^2-a^2+bc-bd= b(c-d)=0

c=d

sher mohammad

b.tech, iit delhi

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Under what conditions the function y = (ax+b)/(cx+d), (ad - bc) = 0 is it’s own inverse?

Under what conditions the function
y = (ax+b)/(cx+d), (ad - bc) = 0 is it’s own inverse?

2 Answers

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