2.5.33. Prove that the image of the function f: R → R , f (x) x2 is the interval [0 ,1) . Do not use calculus. 1+ x2

3. Let f: ZZ be defined by f(x) = 2x² + x. Is f surjective? Is f injective? Justify your answers.

Let n be a positive integer and f1(x), f2(x),f3(x),...fn(x) be differentiable functions. Conjecture what (f1f2f3...fn)' is equal to

Please answer with full explanation and proofs

Study the continuity of the functions in the points mentioned

Show that there exists an x in the interval (0,1) that satisfies the given equation V = 1- x Hint: Create a function f(x) so that you can apply the Intermediate Value Theorem. MacBook Pro esc く 23 24 & 2 3 4 6 7 8 9 - Q W E R Y { S D G つ K く C V M option command command option + I" .. .- リ コ エ リ