Find the domain and range of each of the following real valued functio

1.	Find the domain and range of each of the following real valued functions : f(x)= (ax+b)/(bx-a)f(x) = \(\frac{ax+b}{bx-a}\)
Answer» f(x) = (ax+b)/(bx-a) Clearly, f(x) is defined for all real values of x, Except for the case when bx – a = 0 or x = \(\frac{a}{b}\). When x = \(\frac{a}{b}\), f(x) will be undefined as the division result will be indeterminate. Thus, Domain of f = R – \(\{\frac{a}{b}\}\) Let f(x) = y ⇒ ax + b = y(bx – a) ⇒ ax + b = bxy – ay ⇒ ax – bxy = –ay – b ⇒ x(a – by) = –(ay + b) ∴ x = \(-\frac{(ay+b)}{a-by}\) Clearly, when a – by = 0 or y = \(\frac{a}{b}\) , x will be undefined as the division result will be indeterminate. Hence, f(x) cannot take the value \(\frac{a}{b}\). Thus, Range of f = R – \(\{\frac{a}{b}\}\)

Find the domain and range of each of the following real valued functions : f(x)= (ax+b)/(bx-a)f(x) = \(\frac{ax+b}{bx-a}\)

Answer»

f(x) = (ax+b)/(bx-a)

Clearly,

f(x) is defined for all real values of x,

Except for the case when bx – a = 0 or x = \(\frac{a}{b}\).

When x = \(\frac{a}{b}\),

f(x) will be undefined as the division result will be indeterminate.

Thus,

Domain of f = R – \(\{\frac{a}{b}\}\)

Let f(x) = y

⇒ ax + b = y(bx – a)

⇒ ax + b = bxy – ay

⇒ ax – bxy = –ay – b

⇒ x(a – by) = –(ay + b)

∴ x = \(-\frac{(ay+b)}{a-by}\)

Clearly,

when a – by = 0 or y = \(\frac{a}{b}\) ,

x will be undefined as the division result will be indeterminate.

Hence,

f(x) cannot take the value \(\frac{a}{b}\).

Thus,

Range of f = R – \(\{\frac{a}{b}\}\)