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For the functions f(x)=x/x-3 and g(x)=13/xFind the composition f • x and simplify your answer as much as possible.Write the domain using interval (F•g)(x)=Domain of f•g:

For the functions fxxx3 and gx13xFind the composition f x and simplify your answer as much as possibleWrite the domain using interval FgxDomain of fg class=

Respuesta :

Problem statement

[tex]\begin{gathered} f(x)=\frac{x}{x-3\text{ }} \\ \text{and } \\ g(x)=\frac{13}{x} \end{gathered}[/tex]

Part A

Solution

[tex](f.g)(x)=\frac{\frac{13}{x}}{\frac{13}{x}-3}[/tex]

We can now simplify

[tex]\begin{gathered} (f.g)(x)=\frac{\frac{13}{x}}{\frac{13}{x}-\frac{3}{1}} \\ \\ (f.g)(x)=\frac{\frac{13}{x}}{\frac{13-3x}{x}} \\ \text{simplifying} \\ (f.g)(x)=\frac{13}{x}\times\frac{x}{13-3x} \end{gathered}[/tex]

[tex]\begin{gathered} (f.g)(x)=\frac{13}{x}\times\frac{x}{13-3x} \\ x\text{ can divide x we have} \\ \\ (f.g)(x)=\frac{13}{13-3x} \end{gathered}[/tex]

Part B

The Domain

[tex]\begin{gathered} \text{Let set 13-3x=0} \\ 13=3x \\ x=\frac{13}{3} \end{gathered}[/tex]

The domain is

[tex](-\infty,\frac{13}{3})\text{ U (}\frac{13}{3},\infty)[/tex]