1 For each function from R to R given below determine whethe

(1) For each function from R to R given below, determine whether the function is injective,surjective, both, or neither. No justication is necessary.

y = x2, y = x3, y = x(x ? 1)(x + 1), y = arctan x, y = x2 sin x.

Let X and Y be sets. Some important definitions follow: A mapping or function X 4 Y is a rule which assigns a unique element f(x) E y to each element xEX. We call X the source or domain of f, and call Y the target of f The image of f is defined to be the set im(f) = {f(x) : x E X} The function f is surjective or onto if for all y Y, there exists some x E X such that f(x) = y. In this case, we also say \"f is a surjection.\" that r1-2. In this case, we also say \"f is an injection.\" is the identity map on Y and the composition go f is the identity map on X. In this case, we also The function f is injective or one-to-one if whenever f(x) = f(x2) fr x1,x2 E X, it follows The function f is injective or one-to-one if whenever f()(2) for ,2 X, it follows The function f is bijective if there exists a function g : Y X such that the composition fog say \"f is a bijection.\" (1) For each function from R to R given below, determine whether the function is injective, surjective, both, or neither. No justification is necessary. ,-2, y =3, y = x(z-1)(z + 1), y = arctan x, y =2 sin r.

Solution

given functions are

y =x^2

f(1) =1

f(-1) =1

surjection

y= x^3

f(1) = 1

f(-1) = -1

this is surjective

y = x(x-1) (x+1)

f(0) = 0

f(1) =0

f(-1) =0

injective

y= arctanx

f(pi/4) = 1

f(5pi/4) =1

surjective

y=x^2 sinx

both surjective and injective