Question

Show that the function f : R ⟶ R such that f(x) = 3 − 4x is one-one onto and hence bijective.

Answer 1

We have function f(x) = 3 − 4x which is defined on R.

Let x₁, x₂ ∈ R such that x₁ ≠ x₂.

⇒ −4x₁ ≠ −4x₂ (Multiplying both sides by -4)

⇒ 3 − 4x¹ ≠ 3 − 4x² (Adding 3 both sides)

⇒ f(x₁ ) ≠ f(x₂ ).

This implies that each different element having different images under the function f(x).

Hence, the function f(x) is one-one function. (By definition of one-one function)

Let y ∈ R such that y = 3 − 4x.

⇒ x = \(\cfrac{3-y}4\) ∈ R.

Thus, for each y in the codomain R there exists \(\cfrac{3-y}4\) ∈ R such that f\(\left(\cfrac{3-y}4\right)\) = 3 - 4\(\left(\cfrac{3-y}4\right)\) = y.

Hence, function f(x) = 3 − 4x is onto.

Hence, function f(x) = 3 − 4x is bijective.

(Because, one - one onto functions are calling as bijective function.)

Hence, function f(x) = 3 − 4x is one-one and onto and hence, bijective.