Question

PLEASE HELP, I NEED TO BE HELPED WITH THESE QUESTIONS

Answer 1

Answer:

$(f+g)(x)=\sqrt{3x+7}+\sqrt{3x-7}$

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

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

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

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

Step-by-step explanation:

You are given the following:

$f(x)=\sqrt{3x+7} \text{ and } g(x)=\sqrt{3x-7}$

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:

$\sqrt{3x+7}+\sqrt{3x-7}$

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You are given the following:

$f(x)=x^2+3 \text{ and } g(x)=\sqrt{x-2}$

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

$f(g(x))$

$f(\sqrt{x-2})$ 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:

$(\sqrt{x-2})^2+3$

$x-2+3$

$x+1$

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We are given $y=\frac{4}{x^2}+9$ 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(\frac{4}{x^2}){/tex]    I replace g(x) with 4/x^2:

[tex]\frac{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

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