Answer:
Explanation:
Given the function;
[tex]f(x)=\log _4x[/tex]
To be able to plot the graph of the above function, let's choose values for x and determine the corresponding f(x) values;
When x = 1;
[tex]\begin{gathered} f(1)=\log _41 \\ =0 \end{gathered}[/tex]
When x = 2;
[tex]\begin{gathered} f(2)=\log _42 \\ =\frac{1}{2} \\ =0.5 \end{gathered}[/tex]
When x = 4;
[tex]\begin{gathered} f(4)=\log _44 \\ =1 \end{gathered}[/tex]
Let's go ahead and determine the inverse of f(x) following the below steps;
Step 1: Replace f(x) with y;
[tex]y=\log _4x[/tex]
Step 2: Interchange the positions of x and y;
[tex]x=\log _4y[/tex]
Step 3: Solve for y;
[tex]\begin{gathered} 4^x=y \\ y=4^x \end{gathered}[/tex]
Step 4: Replace y with f^-1(x);
[tex]f^{-1}(x)=4^x[/tex]
To graph the above function, let's also choose values for x and determine the corresponding f^-1(x) values;
When x = -4;
[tex]\begin{gathered} f^{-1}(-4)=4^{-4} \\ =\frac{1}{4^4} \\ =\frac{1}{256} \\ =0.004 \end{gathered}[/tex]
When x = 0;
[tex]\begin{gathered} f^{-1}(0)=4^0 \\ =1 \end{gathered}[/tex]
When x = 1;
[tex]\begin{gathered} f^{-1}(1)=4^1=4 \\ \end{gathered}[/tex]
When x = 2;
[tex]f^{-1}(2)=4^2=16[/tex]
See the below graph of the f(x) and f^-1(x);
