Please Show your work so I can follow your logic, understand and follow along. P

Question

Please Show your work so I can follow your logic, understand and follow along. Please avoid skipping steps and compressing them into one step. I am looking to actually learn from this, not just get any answer that I can't verify for myself. This is how I will determine the best answer and award more stars. I will refund my points completely if you give me just an answer alone with no real guidance, or if you do something close to that.

An example of a function that maps UCF students to the set of positive integers is f(x) = the number of UCF courses student x has taken. Come up with a separate idea of a function from everyday life and list a table with five inputs to the function and their five corresponding outputs. Prove that the function f(n) = 2n with the domain n Z is a one-to-one (injective) function, but not an onto (surjective) function. Let f(x) = x2 - 6x + 4 for all real x 3. Determine f 1(x). What is the range of f(x) and the domain of f1(x)? Also, show that if we allowed the domain to be all real values of x; the function f would not be invertible. Let f(x) = (2x + 3)2 and let g(x) = ex Determine the functions f(g(x)) and g(f(x)).

Explanation / Answer

a)f(x)= number of alphabets in his name.

f(karan)=5

f(raju)=4

f(avi)=3

f(ramesh)=6

f(dj)=2

b)The function is one-one because

Let us take f(n_1)=f(n_2)

so 2n_1=2n_2

so,n_1=n_2

Therefore it is one-one.

its nor surjective because every point in Z goesnot have a preimage.

For example 1 doesnot have a preimage.

We cannot find n belonging to Z such that f(n)=1.

c)

y=x^2 -6x + 4

= (x-3)^{2} - 5

so f^(-1)(x)=+sqrt((y+5))+3 (we choose + so as to satisfy the domain of f(x))

f(x)=x^2 -6x + 4

= (x-3)^{2} - 5

(x-3)^{2} goes from 0 to infinity.

So, (x-3)^{2} - 5 varies form -5 to infinity.

So range of f is (-5,infinity)

So domain of f^{-1} is (-5,infinity)

If the domain of f(x) would have been all of R,then the inverse couldnot be defined as it couldnot be surjective.

Every point in the range would have two preimages.

d) f(g(x))=(2e^{x}+3)^2

g(f(x))=e^((2x+3)^2)

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