Question

Find the functions and simplify

Answer 1

Given the functions f(x) and g(x), as follows:

`f(x)=(1)/(x-4)`

and

`g(x)=(5)/(x)+4`

Now:

(a) f(g(x)) is obtained as follows:

`\begin{gathered} f(x)=(1)/(x-4)\text{ and }g(x)=(5)/(x)+4 \\ \Rightarrow f(g(x))=(1)/(((5)/(x)+4)-4)\text{ } \\ \Rightarrow f(g(x))=(1)/((5)/(x))=(x)/(5) \\ \Rightarrow f(g(x))=(x)/(5) \end{gathered}`

(b) g(f(x)) is obtained as follows:

`\begin{gathered} f(x)=(1)/(x-4)\text{ and }g(x)=(5)/(x)+4 \\ \Rightarrow g(f(x))=(5)/((1)/(x-4))+4=5(x-4)+4 \\ \Rightarrow g(f(x))=5(x-4)+4=5x-20+4=5x-16 \\ \Rightarrow g(f(x))=5x-16 \end{gathered}`