Question

what is the inverse of the logarithmic function $f(x) = \log_{9}x$? $f^{-1}(x) = x^{9}$ $f^{-1}(x) = -\log_{9}x$ $f^{-1}(x) = 9^{x}$ $f^{-1}(x) = \frac{1}{\log_{9}x}$ complete how can you use a point on the graph of $f^{-1}(x) = 9^{x}$ to determine a point on the graph of $f(x) = \log_{9}x$?

Explanation:

To determine a point on the graph of \( f(x)=\log_{9}x \) from a point on the graph of its inverse \( f^{-1}(x) = 9^{x}\), we use the property of inverse functions. If \((a,b)\) is a point on the graph of \( f^{-1}(x)\), then \((b,a)\) is a point on the graph of \( f(x)\). This is because inverse functions are reflections over the line \( y = x\), which swaps the \( x\)- and \( y\)-coordinates of points. For example, if we have a point \((2, 81)\) on \( f^{-1}(x)=9^{x}\) (since \( 9^{2}=81\)), then the corresponding point on \( f(x)=\log_{9}x\) would be \((81, 2)\) because \(\log_{9}81 = 2\) (since \( 9^{2}=81\)).

Answer:

To find a point on \( f(x)=\log_{9}x \) from a point \((a,b)\) on \( f^{-1}(x) = 9^{x}\), swap the \( x\)- and \( y\)-coordinates to get the point \((b,a)\) on \( f(x)=\log_{9}x \).