Question

In the picture I need the answers to the questions.

Answer 1

a)
`\mathbf{ f(g(x))=x+2√(x) +1}`

b)
`\mathbf{f(g(9))=16}`

Explanation:

We are given:

`f(x)=(x-2)^2\\g(x)=√(x) +3`

We need to find
`a) f(g(x))\\b) f(g(9))`

a) First finding:
`f(g(x))`

It can be found by putting the value of g(x) into f(x)

We are given:

`f(x)=(x-2)^2\\Put\:x=g(x)\:i.e. \: x= √(x) +3\\f(g(x))=(√(x) +3-2)^2\\Now\: solving:\\f(g(x))=(√(x) +1)^2\\Using\:formula\:\mathbf{(a+b)^2=a^2+2ab+b^2}\\f(g(x))=(√(x))^2+2(√(x) )(1)+(1)^2\\ f(g(x))=x+2√(x) +1`

SO, we get:
`\mathbf{ f(g(x))=x+2√(x) +1}`

b) Now finding:
`f(g(9))`

It can be found by putting x=9 in f(g(x))

We have:

`f(g(x))=x+2√(x) +1\\Put\:x=9\\f(g(9))=9+2√(9)+1\\We\:know\: √(9)=3\\ f(g(9))=9+2(3)+1\\f(g(9))=9+6+1\\f(g(9))=16`

So, we get:
`\mathbf{f(g(9))=16}`