Question

Let A = [$-$1, 1]. Then, discuss whether the following functions from A to itself are one-one, onto or bijective:

(i) f(x) = $\frac{x}{2}$ (ii) g(x) = |x| (iii) h(x) = x² [NCERT EXEMPLAR]

Answer 1

(i) f : A $\to$ A, given by f(x) = $\frac{x}{2}$

Injection test:

Let x and y be any two elements in the domain (A), such that f(x) = f(y).

f(x) = f(y)

$\frac{x}{2}$ = $\frac{y}{2}$

x = y

So, f is one-one.

Surjection test:

Let y be any element in the co-domain (A), such that f(x) = y for some element x in A (domain)

f(x) = y

$\frac{x}{2}$ = y

x = 2y, which may not be in A.

For example, if y = 1, then

x = 2, which is not in A.

So, f is not onto.

So, f is not bijective.

(ii) g(x) = |x|

Injection test:

Let x and y be any two elements in the domain (A), such that f(x) = f(y).

f(x) = f(y)

|x| = |y|

x = $\pm$y

So, f is not one-one.

Surjection test:

For y = $-$1, there is no value of x in A.

So, f is not onto.

So, f is not bijective.

(iii) h(x) = x²

Injection test:

Let x and y be any two elements in the domain (A), such that f(x) = f(y).

f(x) = f(y)

x² = y²

x = $\pm$y

So, f is not one-one.

Surjection test:

For y = $-$1, there is no value of x in A.

So, f is not onto.

So, f is not bijective.