Question

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

Answer 1

We are given the following function:

$f(x)=\frac{x}{x-8},g(x)=-\frac{1}{x}$

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:

$f(g(x))=\frac{-\frac{1}{x}}{-\frac{1}{x}-8}$

Now, we simplify the fraction:

$f(g(x))=\frac{1}{1+8x}$

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:

$x=-\frac{1}{8}$

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

$D={}\lbrace x\parallel x<-\frac{1}{8},x>-\frac{1}{8}\rbrace$

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:

$g(f(x))=-\frac{1}{\frac{x}{x-8}}$

Now, we simplify the expression:

$g(f(x))=-\frac{x-8}{x}$

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":

$f(f(x))=\frac{\frac{x}{x-8}}{\frac{x}{x-8}-8}$

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

$f(f(x))=\frac{x}{x-8(x-8)}$

Applying the distributive property and adding like terms we get:

$f(f(x))=\frac{x}{x-8x+64}=\frac{x}{-7x+64}$

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:

$x=\frac{64}{7}$

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

$D=\lbrace x\parallel x<\frac{64}{7},x>\frac{64}{7}\text{ \textbraceright}$

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:

$g(g(x))=-\frac{1}{-\frac{1}{x}}$

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.