Question

the function $f(x)=\log_{3}x$ is represented in the table.

$x$	$f(x)$
3	1
9	2
27	3

complete the table below for the functions inverse, $f^{-1}(x)=3^{x}$.

$x$	$f^{-1}(x)$
1	$\square$
2	$\square$
3	$\square$

Explanation:

Step1: For \( x = 0 \)

We use the inverse function \( f^{-1}(x)=3^{x} \). Substitute \( x = 0 \) into the function: \( f^{-1}(0)=3^{0} \). By the rule of exponents, any non - zero number to the power of 0 is 1, so \( 3^{0}=1 \).

Step2: For \( x = 1 \)

Substitute \( x = 1 \) into \( f^{-1}(x)=3^{x} \). We get \( f^{-1}(1)=3^{1} \). And \( 3^{1}=3 \).

Step3: For \( x = 2 \)

Substitute \( x = 2 \) into \( f^{-1}(x)=3^{x} \). Then \( f^{-1}(2)=3^{2} \). Calculating \( 3^{2}=9 \).

Step4: For \( x = 3 \)

Substitute \( x = 3 \) into \( f^{-1}(x)=3^{x} \). We have \( f^{-1}(3)=3^{3} \). And \( 3^{3}=27 \).

Answer:

For \( x = 0 \), \( f^{-1}(0)=1 \); for \( x = 1 \), \( f^{-1}(1)=3 \); for \( x = 2 \), \( f^{-1}(2)=9 \); for \( x = 3 \), \( f^{-1}(3)=27 \). So the completed table values (from top to bottom) are 1, 3, 9, 27.