Question

Use the pair of functions to find f(g(x)) and g(f(x)) . Simplify your answers. f(x)=√x+8 , g(x)=x^2+9

Answer 1

`\begin{gathered} f(g(x))=x^(2)+9+8 \\ g(f(x))=x+16√(x)+73 \end{gathered}`

Step-by-step explanation

Given;

`f\left(x\right)=√(x)+8,\:g\left(x\right)=x^2+9`
`f(g(x))`

By substitution;

`\begin{gathered} g(x)=x^2+9 \\ f(x^2+9)=√(x^2+9)+8 \\ f(g(x))=√(x^2+9)+8 \end{gathered}`

For g(f(x));

By substitution;

`\begin{gathered} \:g=x^2+9 \\ g(f(x))=\left(√(x)+8\right)^2+9 \end{gathered}`

Solving the equation;

`\begin{gathered} \begin{equation*} \left(√(x)+8\right)^2+9 \end{equation*} \\ =\left(√(x)\right)^2+2√(x)\cdot\:8+8^2+9 \\ =x+16√(x)+64+9 \\ =x+16√(x)+73 \end{gathered}`