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Social ScienceSome of these questions are from the following textbook, Discrete Mathematics withApplications, by Susanna S. Epp. 1 Answer the following questions:Define F: NN by the rule F(n) = 2-3n, for all natural numbers n E N. i. Is Fafunction? Why or why not?&ihhe *** 4r152191i. If we modify the codomain of F to be Z (the set of all integers), is F a function? If f is afunction, is it an injective (i.e., one-to-one) function? Is it a surjective function? Brieflyexplain why.PEWAYI44**# 4pDefine G: RR by the rule G(x) = 23x for all real numbers x. Is G surjective?Explain your answer./0. a #,CS Scanned withCamScanner 5) It's possible to compose two functions to get a composite function. For example,let f R R, and f(x) 2x 1 (i.e., function f maps a real number x to a real numbergiven by 2x +1;), and let g: RR, and g(x) = x ^2 (i.e., function g maps any realnumber to its square). Then function (g f) is defined as follows: (gg(f(x)), i.e., function (g f) maps x to g(f(x)) (i.e., x is first mapped to a real numberf(x), and then f(x) is in turn mapped to number g(f(x)) using rule of function g.So (g f)(x) g(2x 1) (2x 1)^2.i. Is (g f)(x) a function? Why?f)(x)ii. Calculate (gf)(1), (g. f)(0.2).Iii. What is function (f g)(x)= f(g(x))? Calculate (fg)(1), (fe)(0.1)
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Some of these questions are from the following textbook, Discrete Mathematics with Applications, by Susanna S. Epp. 1 Answer the following questions: Define F: NN by the rule F(n) = 2-3n, for all natural numbers n E N. i. Is Fa function? Why or why not? &ihhe *** 4r152191 i. If we modify the codomain of F to be Z (the set of all integers), is F a function? If f is a function, is it an injective (i.e., one-to-one) function? Is it a surjective function? Briefly explain why. PEWAY I 44**# 4 p Define G: RR by the rule G(x) = 2 3x for all real numbers x. Is G surjective? Explain your answer. /0. a # , CS Scanned with CamScanner
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5) It's possible to compose two functions to get a composite function. For example, let f R R, and f(x) 2x 1 (i.e., function f maps a real number x to a real number given by 2x +1;), and let g: RR, and g(x) = x ^2 (i.e., function g maps any real number to its square). Then function (g f) is defined as follows: (g g(f(x)), i.e., function (g f) maps x to g(f(x)) (i.e., x is first mapped to a real number f(x), and then f(x) is in turn mapped to number g(f(x)) using rule of function g. So (g f)(x) g(2x 1) (2x 1)^2.i. Is (g f)(x) a function? Why? f)(x) ii. Calculate (g f)(1), (g. f)(0.2). I ii. What is function (f g)(x)= f(g(x))? Calculate (f g)(1), (f e)(0.1)
Expert Answer
4(i)Consider the given mapping F: N→N defined by
F(n)=2-3n , for all natural numbers n belongs to N
for, F to be a function, for each n there will be exactly one value of F(n),
for example: n=1 gives F(n)=-1, n=2 gives F(n)=-4 but -1 and -4 are not natural numbers,
so, for each n there does not exist value of F(n) on the defined codomain.
Hence, F(n) is not a function.
(ii)Now, when the codomain if F is modified to be Z (the set of all integers) , for each n there will be exactly one value of F(n) on defined codomain.
Hence, F(n) is a function.
So, here F:N→Z defined by F(n)=2-3n
Now, check injectivity:
Let n1 and n2 belongs to N such that
F( n1)= F( n2)
2-3n1=2-3n2
n1= n2
Thus, F is one-to-one function.
Now, check surjectivity:
It is known that a function f is surjective if for every y there exist an x such that f(x)=y,
Now, here, let y=F(n)=0 from Z, then
2-3n=0
n=2/3, which is not a natural number.
So, for every y there does not exist an x such that f(x)=y,
Hence, F is not surjective.
(iii)
Now, Consider the given mapping G: R→R defined by
G(x)=2-3x ,
Now, let y=G(x) from R,then
y=2-3x
3x=2-y
x=(2-y)/3
Now, G(x)=G((2-y)/3)
=2-3[(2-y)/3]
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