Question

f(x) = (2x+1)^(1/2)
g is the inverse of f(x)
g'(3) = ?

Answer 1

You can say \(f(x)\) is \(y\), and then swap \(x\) and \(y\) to find the inverse:
\[ (2y+1)^{1/2}=x\\
\left((2y+1)^{1/2}\right)^2= x^2\\
(2y+1)=x^2\\
2y=x^2-1\\
y=\frac{x^2-1}{2}\]
Now, this is your \(g(x)\) function, i.e. \(g(x)=\frac{x^2-1}{2}\). Now you need to find \(g'(x)\)
\[g'(x)=\frac{1}{2}(2x)=x\]

Then, \(g'(3)=3\)