To create the table and graph the inverse function [tex]\( f^{-1}(x) \)[/tex], we will follow these steps:
1. Identify the inverse values [tex]\( f^{-1}(x) \)[/tex] for each provided [tex]\( x \)[/tex]-value using the original function [tex]\( f(x) = 2^x \)[/tex].
2. Plot the points on the coordinate plane.
Firstly, let's understand the given function [tex]\( f(x) = 2^x \)[/tex]. This function models the number of cells [tex]\( f(x) \)[/tex] after [tex]\( x \)[/tex] hours. To find the inverse [tex]\( f^{-1}(x) \)[/tex], we need to solve [tex]\( 2^x = y \)[/tex] for [tex]\( x \)[/tex], which means [tex]\( x = \log_2(y) \)[/tex].
Let's now calculate [tex]\( f^{-1}(x) \)[/tex] for each provided value in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f^{-1}(x) \\
\hline
1 & \log_2(1) = 0 \\
2 & \log_2(2) = 1 \\
4 & \log_2(4) = 2 \\
8 & \log_2(8) = 3 \\
16 & \log_2(16) = 4 \\
\hline
\end{array}
\][/tex]
Thus, the calculated points for the inverse function [tex]\( f^{-1}(x) \)[/tex] are:
- [tex]\( (1, 0) \)[/tex]
- [tex]\( (2, 1) \)[/tex]
- [tex]\( (4, 2) \)[/tex]
- [tex]\( (8, 3) \)[/tex]
- [tex]\( (16, 4) \)[/tex]
Next, we can plot these points on a coordinate plane to graph [tex]\( f^{-1}(x) \)[/tex]. Note that since this is the inverse function, for each point [tex]\( (x, f(x)) \)[/tex] on the graph of [tex]\( f \)[/tex], there is a corresponding point [tex]\( (f(x), x) \)[/tex] on the graph of [tex]\( f^{-1} \)[/tex].
So, the points to plot for the graph of [tex]\( f^{-1}(x) \)[/tex] are:
- [tex]\((1, 0)\)[/tex]
- [tex]\((2, 1)\)[/tex]
- [tex]\((4, 2)\)[/tex]
- [tex]\((8, 3)\)[/tex]
- [tex]\((16, 4)\)[/tex]
To summarize:
- Mark these points on the coordinate plane.
- The resulting graph of the inverse function will show the relationship between the number of cells and the time taken to reach that number, essentially reversing the original function's axes.
This concludes the graphical representation of the inverse function based on the provided values.
