Question

Let f(x)=log square root of x to base of 3.
Show that f^-1(x)=3^2x.
Write down the range of the inverse of f(x)

Answer 1

\[f(x) = y = \log_3{\sqrt{x}}\]

to find the inverse of a function, we would swap x and y in the function and the put it back in the form of f(x).
giving us:
\[x = \log_3{\sqrt{y}}\]

for the general case:
\[z = \log_b (d) \rightarrow d = b^{z}\]

so in the inverse of the function

\[\sqrt{y} = 3^x \rightarrow y = (3^x)^2 = 3^{2x}\]

let me know if you have any questions ^_^

Answer 2

so how would i find the range of the inverse?