f(x)=1/x-5, g(x)=5x-1/x A. Use composition to prove whether or not the functions are inverses of each other. B. Express the domain of the compositions using interval notation.

Answer:Not inverse of each otherDomain : [-∞,0) U (0,5) U (5,∞]Step-by-step explanation:Given in the question two functionsf(x)=1/x-5g(x)=5x-1/xTo find that each of them are inverse of each other we will use compositionf(g(x))[tex]\frac{1}{\frac{5x-1}{x}-5 }[/tex]take LCM[tex]\frac{1}{\frac{5x-1-5x}{x}}[/tex]5x will be cancel[tex]\frac{1}{\frac{-1}{x}}[/tex]1 ÷ (-1/x)1 × (-x/1)-xNow,g(f(x))[tex]\frac{5\frac{1}{x-5} -1}{\frac{1}{x-5}}[/tex][tex]\frac{5}{x-5} -1}[/tex] × [tex]5-x[/tex][tex]\frac{5-x+5}{x-5} * (x-5)[/tex]10-xAs it ended up with different answers, so f(x) and g(x) are not inverse of each otherThe domain are all the possible x-values of function except x ≠ 0 and x ≠ 5 We can conclude that the domain of the composition function isDomain : [-∞,0) U (0,5) U (5,∞]