Question

Let f(x)= 1/x-2 and g(x)=5/x+2Find the following functions. Simplify your answers.F(g(x))=g(f(x))=

Answer 1

Answer:
`\begin{gathered} a)\text{ }f(g(x))\text{ = }(x)/(5) \\ \\ b)\text{ g\lparen f\lparen x\rparen\rparen = 5x - 8} \end{gathered}`

Step-by-step explanation:

Given:

`\begin{gathered} f(x)\text{ = }\frac{1}{x\text{ - 2}} \\ g(x)\text{ = }(5)/(x)\text{ + 2} \end{gathered}`

To find:

a) f(g(x)) b) g(f(x))

`\begin{gathered} a)\text{ f\lparen g\lparen x\rparen\rparen: we will substitue x in f\lparen x\rparen with g\lparen x\rparen} \\ f(g(x))\text{ = }(1)/(((5)/(x)+2)-2) \\ \\ f(g(x))\text{ = }(1)/(((5+2x)/(x))-2) \\ \\ f(g(x))\text{ = }(1)/(((5+2x-2x)/(x)))\text{ = }(1)/((5)/(x)) \\ \\ f(g(x))\text{ = }(x)/(5) \end{gathered}`
`\begin{gathered} b)\text{ g\lparen f\lparen x\rparen\rparen: we will substitue x in g\lparen x\rparen with f\lparen x\rparen} \\ g(f(x))\text{ = }(5)/((1)/(x-2))+2 \\ \\ g(f(x))\text{ = }\frac{5(x\text{ -2\rparen}}{1}+2 \\ \\ g(f(x))\text{ = }5(x\text{ -2\rparen}+2\text{ = 5x - 10 + 2} \\ \\ g(f(x))\text{ = 5x - 8} \end{gathered}`