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Under what conditions the function y = (ax+b)/(cx+d), (ad - bc) = 0 is it’s own inverse?
First inverse the function :x = ay + b / cy + dxcy + dx = ay + by = b - dx/ cx - aIt should be equal to original functionb - dx/ cx - a = ax + b/cx + dbcx + bd - dcx^2 - d^2x = acx^2 + bcx - a^2x - abBy this, we get a = -d should be the other condition.ThanksBharataskiitians facultyIIT Delhi
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 + bcxy + dx = ay + bcxy - ay = b - dxy(cx - a) = b - dxy = (b - dx) / (cx - a) # inversefor 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)implyc(a+d)=0a=-dd^2-a^2+bc-bd= b(c-d)=0c=dsher mohammadb.tech, iit delhi
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