A mapping f: R → R defined by f (x) = ex where x ∈ R is (a) one-o

1.	A mapping f: R → R defined by f (x) = ex where x ∈ R is (a) one-one onto (b) one-one into (c) many-one into (d) many-one into
Answer» Answer: (b) one - one into For all x₁, x₂ ∈ R, f (x₁) = f (x₂) ⇒ e^x₁ = e^x₂ ⇒ x₁ = x₂ ⇒ f is one-one. Let y = f (x) = e^x ⇒ x = log_ey Now x is not defined for y < 0 ∴ y ∈ R does not have a pre-image in R when y is negative. ∴ f is into.

A mapping f: R → R defined by f (x) = ex where x ∈ R is (a) one-one onto (b) one-one into (c) many-one into (d) many-one into

Answer»

Answer: (b) one - one into

For all x₁, x₂ ∈ R,

f (x₁) = f (x₂)

⇒ e^x₁ = e^x₂ ⇒ x₁ = x₂

⇒ f is one-one.

Let y = f (x) = e^x

⇒ x = log_ey

Now x is not defined for y < 0

∴ y ∈ R does not have a pre-image in R when y is negative.

∴ f is into.