Let's break down the function [tex]\( f(x) = 2^x - 7 \)[/tex] and evaluate it at the given domain points: [tex]\( x = 0, 1, 2, 3, 4 \)[/tex]. We'll fill in the table with these values and then explain how to plot them on a graph.
### Step-by-Step Evaluation:
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2^0 - 7 = 1 - 7 = -6
\][/tex]
2. For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2^1 - 7 = 2 - 7 = -5
\][/tex]
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2^2 - 7 = 4 - 7 = -3
\][/tex]
4. For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 2^3 - 7 = 8 - 7 = 1
\][/tex]
5. For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 2^4 - 7 = 16 - 7 = 9
\][/tex]
### Function Table Completion:
Let's fill the table with these values:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$f(x)$ & -6 & -5 & -3 & 1 & 9 \\
\hline
\end{tabular}
\][/tex]
### Plotting the Points:
Now we want to plot these points [tex]\((x, f(x))\)[/tex] on a graph. Here are the points we have:
- [tex]\((0, -6)\)[/tex]
- [tex]\((1, -5)\)[/tex]
- [tex]\((2, -3)\)[/tex]
- [tex]\((3, 1)\)[/tex]
- [tex]\((4, 9)\)[/tex]
Steps to Plot:
1. Draw the axes: Draw a Cartesian coordinate system with the x-axis (horizontal) ranging from at least 0 to 4 and the y-axis (vertical) covering at least from -6 to 9.
2. Plot the points:
- For [tex]\( x=0 \)[/tex], plot the point at [tex]\( (0, -6) \)[/tex].
- For [tex]\( x=1 \)[/tex], plot the point at [tex]\( (1, -5) \)[/tex].
- For [tex]\( x=2 \)[/tex], plot the point at [tex]\( (2, -3) \)[/tex].
- For [tex]\( x=3 \)[/tex], plot the point at [tex]\( (3, 1) \)[/tex].
- For [tex]\( x=4 \)[/tex], plot the point at [tex]\( (4, 9) \)[/tex].
### Connecting the points (optional):
If a smooth curve or a line is desired to represent [tex]\( f(x) = 2^x - 7 \)[/tex], make sure to sketch in such a way that it connects the plotted points in a smooth manner that represents the exponential nature of the function [tex]\( 2^x \)[/tex].
This table and the points plotted on the graph visually describe how the function [tex]\( f(x) = 2^x - 7 \)[/tex] behaves across the given domain.
