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

• f(x) = x/2
• g(x) = |x|
• h(x) = x²

To determine whether the functions defined from the set A = [-1, 1] to itself are one-one, onto, or bijective, we will analyze each function individually.

Function f(x) = x/2

This function takes any value in A and divides it by 2.

• One-One: Yes, f is one-one. If f(a) = f(b), then a/2 = b/2 implies a = b.
• Onto: No, f is not onto. The range of f is [-0.5, 0.5], which does not cover the entire set A.
• Bijective: Since f is not onto, it cannot be bijective.

Function g(x) = |x|

This function returns the absolute value of x.

• One-One: No, g is not one-one. For example, g(-1) = g(1) = 1, so different inputs yield the same output.
• Onto: Yes, g is onto. The range is [0, 1], which includes all values in A.
• Bijective: Since g is not one-one, it cannot be bijective.

Function h(x) = x²

This function squares the input value.

• One-One: No, h is not one-one. For instance, h(-1) = h(1) = 1, indicating multiple inputs can produce the same output.
• Onto: Yes, h is onto. The range is [0, 1], which covers all non-negative values in A.
• Bijective: Since h is not one-one, it cannot be bijective.

In summary, f(x) is one-one but not onto, g(x) is onto but not one-one, and h(x) is onto but not one-one. None of the functions are bijective.