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Please solve, fill in the blank for each (a) through (d)

Please solve, fill in the blank for each (a) through (d)-example-1
User Double AA
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We are given the following function:


f(x)=(x)/(x-8),g(x)=-(1)/(x)

Part A. We are asked to determine the following:


f\circ g

This means the composition of the function "f" and "g". To do that we will use the following equivalence:


(f\circ g)(x)=f(g(x))

This means that we will substitute the value of "x" from function "f" by the function "g", like this:


f(g(x))=(-(1)/(x))/(-(1)/(x)-8)

Now, we simplify the fraction:


f(g(x))=(1)/(1+8x)

To determine the domain we must set the denominator to zero to determine the values of "x" for which the composition of the functions is undetermined:


1+8x=0

Now, we solve for "x". First, we subtract 1 from both sides:


8x=-1

Now, we divide both sides by 8:


x=-(1)/(8)

Therefore, the composition is undetermined at the point "x = -1/8". Therefore, the domain is:


D={}\lbrace x\parallel x<-(1)/(8),x>-(1)/(8)\rbrace

Part B. we are asked to determine the following:


g\circ f

This is equivalent to:


(g\circ f)(x)=g(f(x))

This means that we will substitute the value of "x" from function "g" for the value of function "f", like this:


g(f(x))=-(1)/((x)/(x-8))

Now, we simplify the expression:


g(f(x))=-(x-8)/(x)

To determine the domain we set the denominator to zero:


x=0

Therefore, the function is undetermined at "x = 0". This means that the domain is the values of "x" that do not include the value of "x = 0":


D=\lbrace x\parallel x>0,x<0\rbrace

Part C. We are asked to determine:


f\circ f

This is equivalent to:


(f\circ f)(x)=f(f(x))

Now, we substitute the value of "x":


f(f(x))=((x)/(x-8))/((x)/(x-8)-8)

Now, we multiply the numerator and denominator by "x - 8":


f(f(x))=(x)/(x-8(x-8))

Applying the distributive property and adding like terms we get:


f(f(x))=(x)/(x-8x+64)=(x)/(-7x+64)

Now, we set the denominator to zero to determine the domain:


-7x+64=0

Now, we subtract 64 from both sides:


-7x=-64

Now, we divide both sides by -7:


x=(64)/(7)

Therefore, the domain does not include the value of "x = 64/7". Therefore, the domain is:


D=\lbrace x\parallel x<(64)/(7),x>(64)/(7)\text{ \textbraceright}

Part D. We are asked to determine:


(g\circ g)(x)=g(g(x))

Substituting the value of "g(x)" in "g(x)" we get:


g(g(x))=-(1)/(-(1)/(x))

Simplifying we get:


g(g(x))=x

To determine the domain we need to have into account that the function is a polynomial and therefore, is not undetermined at any point. This means that the domain is all the real numbers.

User Mottalrd
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