For each of the following functions, determine whether the function is: Injective (one-to-one). Surjective (onto). ⚫ Bijective. Justify your answers. 1.2 f:ZZ → ZZ such that f(x)=5+x.
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- Which functions are one-to-one? Which functions are onto? Describe the inversefunction for any bijective function.(a) f : Z → N where f is defined by f (x) = x4 + 1(b) f : N → N where f is defined by f (x) = { x/2 if x is even, x + 1 if x is odd}(c) f : N → N where f is defined by f (x) = { x + 1 if x is even, x − 1 if x is odd}Determine whether each of the following functions f : {a,b,c,d} -> {a,b,c,d} is one-to-one and/or onto. (a) f(a) = b, f(b) = a, f(c) = b, f(d) = c (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c (c) f(a) = b, f(b) = a, f(c) = c, f(d) = d (d) f(a) = d, f(b) = a, f(c) = c, f(d) = b (e) f(a) = c, f(b) = d, f(c) = aLet A = {1, 2, 3, 4} and B = {a, b, c}. Give an example of a function f: A -> B that is neither injective nor surjective.
- Determine if each function is injective, surjective or bijective. Give one counterexample for each that it is not. 1. function f from {a,b,c,d} to itself, where f(a) = d, f(b) = b, f(c) = a, f(d) = cpick multiple answers on the second one !For the first problem, in parts b and c, how are the variables being taken from functions that they are not defined in? To clarify, in part c, how is the variable "b" taken from fun2 when it is not defined in that in function. Shouldn't the answer there just be "b(fun1)"
- how many functions a) map {1,2,3.4.5.6} to {up,down,left,right} b) how many functions in part a are subjective? c) how many have f- inverse(up) =2Question 7 The function f(x)=x2 from set of positive real numbers to positive real numbers is not injective not surjective both injective and surjective Neither injective nor surjectiveChoose all that apply: 1. For the functions, n", and c", what is the asymptotic relationship between these functions? Assume that k >= 1 and c > 1 are constants. a. nk is O(c") b. nk is N(c") c. nk is 0(c") 2. For the functions, Ign and logan what is the asymptotic relationship between these functions? a. Ign is O( logsn) b. Ign is N( logsn) c. Ign is 0( logsn)
- Define a function make_derivative that returns a function: the derivative of a function f. Assuming that f is a single- variable mathematical function, its derivative will also be a single-variable function. When called with a number a, derivative will estimate the slope of f at point (a, f(a)). the Recall that the formula for finding the derivative of f at point a is: f'(a) = lim h→0 where h approaches 0. We will approximate the derivative by choosing a very small value for h. The closer h is to 0, the better the estimate of the derivative will be. def make_derivative (f): """Returns a function that approximates the derivative of f. >>> def square (x): f(a+h)-f(a) h Recall that f'(a) = (f(a + h) - f(a)) / h as h approaches 0. We will approximate the derivative by choosing a very small value for h. # equivalent to: square = lambda x: x*X return x*x >>> derivative = make_derivative (square) >>> result = derivative (3) >>> round (result, 3) # approximately 2*3 6.0 |||||| h=0.00001 "***…Answer the following multiple choice questions: - If B represented the set of Boolean values true and false, then the set of Boolean functions B x B → B is... A) finite B) uncountably infinite C) countably infinite - The set of functions from {0,1} to N is... A) finite B) uncountably infinite C) countably infinite - The set of functions from N to {0,1} is... A) finite B) uncountably infinite C) countably infiniteQuestion 11 For all a and b in the domain of a function f, the function is injective iff f(a)=f(b)→a3b f(a)=f(b) a b f(a) #f(b) a=b y)=x iff f(x)=y Moving to another question will save this response. o search