Question

PLEASE HELP, I NEED TO BE HELPED WITH THESE QUESTIONS

Answer 1

`(f+g)(x)=√(3x+7)+√(3x-7)`

`f(g(x))=x+1`

`f(x)=x+9 \text{ and } g(x)=(4)/(x^2)`

`f^(-1)(x)=\frax{x+2}{3}`

Let me know if you have any questions about any of my work.

Explanation:

You are given the following:

`f(x)=√(3x+7) \text{ and } g(x)=√(3x-7)`

and asked to find
`(f+g)(x) \text{ which means } f(x)+g(x)`.

If you add those because we are asked to find f(x)+g(x) you get:

`√(3x+7)+√(3x-7)`

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You are given the following:

`f(x)=x^2+3 \text{ and } g(x)=√(x-2)`

and asked to find
`f(g(x))`.

`f(g(x))`

`f(√(x-2))` I replaced g(x) with sqrt(x-2) because that is what it equals.

Now this last thing means to replace old input in x^2+3 with new input sqrt(x-2) giving us:

`(√(x-2))^2+3`

`x-2+3`

`x+1`

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We are given
`y=(4)/(x^2)+9` and asked to find g(x) and f(x) such that y=f(g(x)).

We have choices so let's use the choices:

Choice A:

`f(g(x))`

`f((4)/(x^2)){/tex]    I replace g(x) with 4/x^2:</p><p>[tex](4)/(x^2)+9` I replaced the old input x with new input 4/x^2.

This was actually the desired result.

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To find the inverse of f(x)=3x-2 or y=3x-2, your objective is to swap x and y and then remake y the subject.

y=3x-2

Swap x and y:

x=3y-2

Now solve for y.

Add 2 on both sides:

x+2=3y

Divide both sides by 3:

(x+2)/3=y

y=(x+2)/3

`f^(-1)(x)=\frax{x+2}{3}`