Question

Question 45.Graph each function and its inverse on the same set of axis. Label two points on graph.

Answer 1

Step-by-step explanation:

Given the function;

`f(x)=\log _4x`

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;

`\begin{gathered} f(1)=\log _41 \\ =0 \end{gathered}`

When x = 2;

`\begin{gathered} f(2)=\log _42 \\ =(1)/(2) \\ =0.5 \end{gathered}`

When x = 4;

`\begin{gathered} f(4)=\log _44 \\ =1 \end{gathered}`

Let's go ahead and determine the inverse of f(x) following the below steps;

Step 1: Replace f(x) with y;

`y=\log _4x`

Step 2: Interchange the positions of x and y;

`x=\log _4y`

Step 3: Solve for y;

`\begin{gathered} 4^x=y \\ y=4^x \end{gathered}`

Step 4: Replace y with f^-1(x);

`f^(-1)(x)=4^x`

To graph the above function, let's also choose values for x and determine the corresponding f^-1(x) values;

When x = -4;

`\begin{gathered} f^(-1)(-4)=4^(-4) \\ =(1)/(4^4) \\ =(1)/(256) \\ =0.004 \end{gathered}`

When x = 0;

`\begin{gathered} f^(-1)(0)=4^0 \\ =1 \end{gathered}`

When x = 1;

`\begin{gathered} f^(-1)(1)=4^1=4 \\ \end{gathered}`

When x = 2;

`f^(-1)(2)=4^2=16`

See the below graph of the f(x) and f^-1(x);