Find whether the following functions are one-one or many-one?(i) f: R

1.	Find whether the following functions are one-one or many-one?(i) f: R – {3} → R defined by f (x) = \(\frac{5x+7}{x-3}\) , x ∈ R − {3} (ii) Modulus function f : R → R defined by f (x) = \(\ f(n) =\begin{cases}x \,; & \quad x \geq 0\\-x \,; & \quad x <0\end{cases}\)(iii) Greatest integer function f : R → R defined by f (x) = [x] = n (an integer) for all n ≤ x ≤ n + 1.
Answer» Given: f (x) = \(\frac{5x+7}{x-3}\) , x ∈ R - {3} Let a₁, a₂ be two arbitrary elements ∈R – {3} such that f (a₁) = f (a₂). Then, f (a₁) = f(a₂) ⇒ \(\frac{5a_1+7}{a_1-3} = \frac{5a_2+7}{a_2-3}\) ⇒ 5a₁a₂ + 7a₂ – 15a₁ – 21 = 5a₂a₁ – 21 + 7a₁ – 15a₂ ⇒ – 22a₁ = – 22a₂ ⇒ a₁ = a₂ ⇒ f is one-one. (ii) Given: f (x) = \| x \| ⇒ f (1) = \| 1 \| = 1 and f (– 1) = \|– 1\| = 1 Thus, f (1)= f (– 1) = 1 but 1 ≠ – 1 ∴ f is many-one. (iii) ∵ f (x) = [x] = n for all n ≤ x < n + 1, so, 1 ≤ x < 2 ⇒ f (x) = 1 ∴ The elements 1.1, 1.25, 1.4, 1.9 .... ∈ domain are mapped onto to the same value 1 in the range. Hence the function f is many-one.

Find whether the following functions are one-one or many-one?(i) f: R – {3} → R defined by f (x) = \(\frac{5x+7}{x-3}\) , x ∈ R − {3} (ii) Modulus function f : R → R defined by f (x) = \(\ f(n) =\begin{cases}x \,; & \quad x \geq 0\\-x \,; & \quad x <0\end{cases}\)(iii) Greatest integer function f : R → R defined by f (x) = [x] = n (an integer) for all n ≤ x ≤ n + 1.

Answer»

Given:

f (x) = \(\frac{5x+7}{x-3}\) , x ∈ R - {3}

Let a₁, a₂ be two arbitrary elements ∈R – {3} such that f (a₁) = f (a₂).

Then, f (a₁) = f(a₂) ⇒ \(\frac{5a_1+7}{a_1-3} = \frac{5a_2+7}{a_2-3}\)

⇒ 5a₁a₂ + 7a₂ – 15a₁ – 21 = 5a₂a₁ – 21 + 7a₁ – 15a₂

⇒ – 22a₁ = – 22a₂

⇒ a₁ = a₂

⇒ f is one-one.

(ii) Given: f (x) = | x |

⇒ f (1) = | 1 | = 1 and f (– 1) = |– 1| = 1

Thus, f (1)= f (– 1) = 1 but 1 ≠ – 1

∴ f is many-one.

(iii) ∵ f (x) = [x] = n for all n ≤ x < n + 1, so, 1 ≤ x < 2 ⇒ f (x) = 1

∴ The elements 1.1, 1.25, 1.4, 1.9 .... ∈ domain are mapped onto to the same value 1 in the range.

Hence the function f is many-one.