Question

Use the pair of functions to find f(g(x)) and g(f (x)). Simplify your answers.f(x) = square root(x)+6, g(x) = x^2 + 9

Answer 1

The solution for the function of function are

f(g(x)) = √(x² + 3²) + 6

g(f (x)) = x + 12√(x) + 45

How to solve the function

Information given in the problem

f(x) = √(x) + 6

g(x) = x² + 9

Solving f(g(x))

f(x) = √(x) + 6

f(g(x)) = √(x² + 9) + 6

f(g(x)) = √(x² + 3²) + 6

Solving for g(f (x))

g(x) = x² + 9

g(f (x))) = (√(x) + 6)² + 9

g(f (x)) = (√(x) + 6) (√(x) + 6) + 9

g(f (x)) = x + 12√(x) + 36 + 9

g(f (x)) = x + 12√(x) + 45

Answer 2

`\begin{gathered} f(g(x))\text{ = }\sqrt[]{x^2+9}\text{ + 6} \\ \\ g(f(x))\text{ = x }+12\text{ }\sqrt[]{x}\text{ + 45} \end{gathered}`Step-by-step explanation:
`\begin{gathered} \text{Given:} \\ f(x)=\text{ }\sqrt[]{x}\text{ + 6} \\ g(x)=x^2+\text{ 9} \end{gathered}`

To get f(g(x)): we will substitute the x in f(x) with the function g(x):

`\begin{gathered} f(g(x))\text{ = }\sqrt[]{(x^2+9)}\text{ + 6} \\ f(g(x))\text{ = }\sqrt[]{x^2+9}\text{ + 6} \\ \text{ (it can't be simplified further)} \end{gathered}`

To get g(f(x)): we will substitute the x in g(x) with function f(x):

`\begin{gathered} g(f(x))\text{ = (}\sqrt[]{x}+6)^2\text{ + 9} \\ g(f(x))\text{ = (}\sqrt[]{x}+6)\text{(}\sqrt[]{x}+6)\text{ + 9} \\ g(f(x))\text{ = (}\sqrt[]{x})\text{(}\sqrt[]{x}+6)\text{ }+6\text{ (}\sqrt[]{x}+6)\text{ + 9} \end{gathered}`
`\begin{gathered} g(f(x))\text{ = x }+6\text{ }\sqrt[]{x}\text{ }+6\text{ }\sqrt[]{x}\text{ + 36 + 9} \\ g(f(x))\text{ = x }+12\text{ }\sqrt[]{x}\text{ + 45} \end{gathered}`