Question

Find whether the following functions are one-one.

(i) f : R → R defined by f(x) = x² + 5

(ii) f : R – {3} → R defined by f(x) = \(\frac{5x+7}{x-3}\) for x ∈ R – {3}

Answer 1

(i) f: R → R, defined by f(x) = x² + 5

Note that f(-x) = f(x) = x² + 5

∴ f is not one-one (i.e., many-one) function.

(ii) f: R – {3} → R, defined by f(x) = \(\frac{5x+7}{x-3}\)

Let f(x₁) = f(x₂)

\(\therefore\frac{5x_1+7}{x_1-3}=\frac{5x_2+7}{x_2-3}\)

∴ 5x₁ x₂ – 15x₁ + 7x₂ – 21 = 5x₁ x₂ – 15x₂ + 7x₁ – 21

∴ 22(x₁ – x₂ ) = 0

∴ x₁ = x₂

∴ f is a one-one function