Question

Consider the reciprocal function

f: ℝ − {0} → ℝ − {0} defined by f(x) =

1
/x
(a) Show that f is bijective?

(b) Is f bijective if the domain ℝ -{0} is replaced by ℕ​

Answer 1

Consider the reciprocal function, f : ℝ − {0} → ℝ − {0} defined by f(x) = 1/x

we have to show that f is bijective function.

also we have to check f is bijective or not if the domain ℝ-{0} is replaced by ℕ.

solution : function, f(x) = 1/x

differentiating both sides we get,

f'(x) = -1/x²

for all value of ℝ , f'(x) < 0 it means function is strictly decreasing. it shows that function is one - one function.

now let's check function is onto or not.

let y = f(x) = 1/x

⇒x = f(y) = 1/y

domain of f(y) = range of f(x) = ℝ - {0}

so co - domain = Range

Therefore the f is onto function.

here f is one - one and onto then f is definitely bijective function.

now domain ℝ - {0} is replaced by ℕ.

i.e., f : ℕ → ℝ - {0}

here range of function, f(x) = 1/x is also ℕ.

here range ≠ co - domain

so f is not onto function. Therefore f is not bijective function if domain ℝ - {0} is replaced by ℕ.