is the function defined or not defined Recall x is the floor

is the function defined or not defined

(Recall [x] is the floor function - the largest integer less than or equal to x.) f: [0, infinity) [0, infinity); f(x) = ln(x+ 1) g : R R; g(x) = 2x +5 Is f^-1 defined? If so, what is it? If not, why not? Is g^-1 defined? If so, what is it? If not, why not? Is f o g defined? If so, what is it? If not, why not? Is g o f defined? If so, what is it? If not, why not?

Solution

yes inverse of f(x) is defined.

to find the inverse of f(x) substitute x in place of y and y in place of x

ln(y+1)=x

y+1=e^x

y=e^x-1

f^-1(x)=e^x-1

b)yes inverse of g is defined

g^-1 =(x-5)/2

c)f(g(x))-put the value of g(x) in x of f(x)

f0g-->ln(2x+6) is defined when 2x+6>0 ,x>-3

d)gof

put the value of f(x) in x of gx

g(ln(x+1))=2(ln(x+1)+5