Answers:
(i)
f(x) is defined for all real values of x, except when bx – a = 0 or x = a/b.
Domain (f) = R – (a/b)
Consider,
f (x) = y
(ax+b)/(bx-a) = y
ax + b = y(bx – a)
ax + b = bxy – ay
ax – bxy = –ay – b
x(a – by) = –(ay + b)
∴ x = – (ay+b)/(a-by)
If, a – by = 0 or y = a/b
f(x) cannot take the value a/b.
∴ Range (f) = R – (a/b)
(ii)
f(x) is defined for all real values of x, except when cx – d = 0 or x = d/c. Domain (f) = R – (d/c)
Consider, f(x) = y
(ax-b)/(cx-d) = y
ax – b = y(cx – d)
ax – b = cxy – dy
ax – cxy = b – dy
x(a – cy) = b – dy
∴ x = (b-dy)/(a-cy)
a – cy = 0 or y = a/c,
f(x) cannot take the value a/c.
∴ Range (f) = R – (a/c)