what is the inverse of the logarithmic function $f(x)=log_9x$?\n$f^{-1}(x)=x^9$ $f^{-1}(x)=-log_9x$\n$f^{-1}(…

what is the inverse of the logarithmic function $f(x)=log_9x$?\n$f^{-1}(x)=x^9$ $f^{-1}(x)=-log_9x$\n$f^{-1}(x)=9^x$ $f^{-1}(x)=\frac{1}{log_9x}$\ncomplete\nhow can you use a point on the graph of $f^{-1}(x)=9^x$ to determine a point on the graph of $f(x)=log_9x$?\nswitch the x - and y - coordinates.\nuse the opposites of the x - and y - coordinates.\nuse the reciprocals of the x - and y - coordinates.

what is the inverse of the logarithmic function $f(x)=log_9x$?\n$f^{-1}(x)=x^9$ $f^{-1}(x)=-log_9x$\n$f^{-1}(x)=9^x$ $f^{-1}(x)=\frac{1}{log_9x}$\ncomplete\nhow can you use a point on the graph of $f^{-1}(x)=9^x$ to determine a point on the graph of $f(x)=log_9x$?\nswitch the x - and y - coordinates.\nuse the opposites of the x - and y - coordinates.\nuse the reciprocals of the x - and y - coordinates.

Answer

Explanation:

Step1: Recall inverse - function property

If (y = f(x)) and (x = f^{-1}(y)), for a logarithmic function (y=\log_{a}x) (where (a>0,a\neq1)), its inverse is found by converting the logarithmic equation to an exponential equation. The general form of converting (\log_{a}x = y) to exponential form is (x=a^{y}). For (y = \log_{9}x), the inverse function (x = 9^{y}), which can be written as (y = 9^{x}) when we swap (x) and (y) to get the inverse function in the standard (y = f^{-1}(x)) form.

Step2: Recall relationship between points on a function and its inverse

The graphs of a function (y = f(x)) and its inverse (y = f^{-1}(x)) are symmetric about the line (y=x). If ((x_1,y_1)) is a point on the graph of (y = f^{-1}(x)), then ((y_1,x_1)) is a point on the graph of (y = f(x)). This means we switch the (x -) and (y -)coordinates.

Answer:

  1. (f^{-1}(x)=9^{x})
  2. Switch the x- and y-coordinates.