Question

Please hurry! This is due soon.

Answer 1

Hi!

We can see here that this is a composition question.

And since the composition of g of f of x is x, we can conclude that g(x) is the inverse of f(x) (if you're confused, search up the definition of an inverse function).

To find an inverse function, we can take the f(x) function and change the positions of the x and y variables.

$f(x)=\frac{e^7^x+\sqrt{3}}{2}$

$y=\frac{e^7^x+\sqrt{3}}{2}$

$x=\frac{e^7^y+\sqrt{3}}{2}$

$2x=e^7^y+\sqrt{3}$

$e^7^y=2x-\sqrt{3}$

$7y=ln(2x-\sqrt{3})$

$y=\frac{ln(2x-\sqrt{3})}{7}$

Which is answer choice A, to check your work, you can solve the composition of g(f(x)), which will get you x.

$g(f(x))$

$g(\frac{e^7^x+\sqrt{3}}{2})$

$\frac{ln(2(\frac{e^7^x+\sqrt{3}}{2})-\sqrt{3}}{7}$

2s cancel.

$\frac{ln(e^7^x+\sqrt{3})-\sqrt{3}}{7}$

The natural log and e cancel.

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

$\sqrt{3}$s cancel.

$\frac{7x}{7}$

7s cancel.

$x$

Hope this helps!