A function from $\mathrm{A}=\{x:-1 \leq x \leq 1\}$ to itself which is not a bijection is
A function from $\mathrm{A}=\{x:-1 \leq x \leq 1\}$ to itself which is not a bijection is
$f(x)=x|x|$
$f(x)=x^3$
$f(x)=x^2$
$f(x)=\sin \left(\frac{\pi x}{2}\right)$
Solution
A function that maps the set A={x: -1 ≤ x ≤ 1} to itself and is not a bijection is f(x) = x² because f(-1) = 1 and f(1) = 1, meaning different inputs (-1 and 1) map to the same output (1), making it not injective (one-to-one) and therefore not a bijection.
Why f(x) = x² is not a bijection:
Not Injective (One-to-One): For a function to be injective, each distinct element in the domain must map to a unique element in the codomain.
In this case, the inputs -1 and 1 are distinct elements of set A,
but they both map to the same output, 1 (i.e., f(-1) = (-1)² = 1 and f(1) = (1)² = 1).
Not Surjective (Onto): The range of f(x) = x² for x in [-1, 1] is.
However, the codomain is A = [-1, 1].
Since the range does not cover the entire codomain (for example, the element -0.5 in A is not mapped to by any x in A), the function is not surjective.
A function must be both injective and surjective to be a bijection.
Since f(x) = x² fails to be injective (and also surjective), it is not a bijection.