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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

Use the pair of functions to find f(g(x)) and g(f (x)). Simplify your answers.f(x-example-1

2 Answers

6 votes

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

User Rahin
by
7.8k points
1 vote

\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}

User Santiago Rebella
by
8.2k points

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