Question

For the following function round your answer to two decimal places if necessary

Answer 1

Given the functions f(x) and g(x) defined by:

`\begin{gathered} f(x)=\sqrt{(2x)/(3)} \\ \\ g(x)=√(x^2+15) \end{gathered}`

To find the composite function evaluated at 1, we use the definition of composite functions:

`(f\circ g)(x)=f(g(x))`

Then, for x = 1, we find g(1):

`\begin{gathered} g(1)=√(1^2+15)=√(16) \\ \\ \Rightarrow g(1)=4 \end{gathered}`

Now, we use this result in f(x):

`\begin{gathered} f(4)=\sqrt{(2\cdot4)/(3)}=\sqrt{(8)/(3)} \\ \\ \Rightarrow f(4)\approx1.63 \end{gathered}`

Finally:

`\begin{gathered} (f\circ g)(1)=f(g(1))=f(4) \\ \\ \therefore(f\circ g)(1)=1.63 \end{gathered}`