Let f: R → R be a function defined by f (x) = \(\frac{x-m}{x-n}\) ,

1.	Let f: R → R be a function defined by f (x) = \(\frac{x-m}{x-n}\) , when m ≠ n, then(a) f is one-one onto (b) f is many-one onto (c) f is one-one into (d) f is many-one into
Answer» Answer: (c) = f is one - one into ∀ (x, y) ∈ R, f (x) = f (y) ⇒ \(\frac{x-m}{x-n}\) = \(\frac{y-m}{y-n}\) ⇒ (x – m) (y – n) = (y – m) (x – n) ⇒ xy – my – nx + mn = yx – mx – ny + mn ⇒ mx – nx = my – ny ⇒ (m – n) x = (m – n) y ⇒ x = y ⇒ f is one-one. Let z = f (x) = \(\frac{x-m}{x-n}\) ⇒ zx – zn = x – m ⇒ zx – x = zn – m ⇒ x (z – 1) = zn – m ⇒ x = \(\frac{zn-m}{z-1} = \frac{m-zn}{1-z}\) x is not defined for z = 1 ⇒ for z = 1, there exists no pre-image in R ⇒ f is not onto. ∴ f is one-one, into function.

Let f: R → R be a function defined by f (x) = \(\frac{x-m}{x-n}\) , when m ≠ n, then(a) f is one-one onto (b) f is many-one onto (c) f is one-one into (d) f is many-one into

Answer»

Answer: (c) = f is one - one into

∀ (x, y) ∈ R, f (x) = f (y)

⇒ \(\frac{x-m}{x-n}\) = \(\frac{y-m}{y-n}\)

⇒ (x – m) (y – n) = (y – m) (x – n)

⇒ xy – my – nx + mn = yx – mx – ny + mn

⇒ mx – nx = my – ny

⇒ (m – n) x = (m – n) y

⇒ x = y

⇒ f is one-one.

Let z = f (x) = \(\frac{x-m}{x-n}\)

⇒ zx – zn = x – m

⇒ zx – x = zn – m

⇒ x (z – 1) = zn – m

⇒ x = \(\frac{zn-m}{z-1} = \frac{m-zn}{1-z}\)

x is not defined for z = 1

⇒ for z = 1, there exists no pre-image in R

⇒ f is not onto.

∴ f is one-one, into function.