Let's go through the process of solving the equation [tex]\( f(x) = 12 \)[/tex] both graphically and algebraically.
### Algebraic Solution
Given the function:
[tex]\[ f(x) = (x+3)^3 + 4 \][/tex]
We need to find [tex]\( x \)[/tex] when [tex]\( f(x) = 12 \)[/tex].
1. Set up the equation:
[tex]\[ (x+3)^3 + 4 = 12 \][/tex]
2. Subtract 4 from both sides:
[tex]\[ (x+3)^3 = 8 \][/tex]
3. Take the cube root of both sides:
[tex]\[ x+3 = 2 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 2 - 3 \][/tex]
[tex]\[ x = -1 \][/tex]
We obtain one real solution: [tex]\( x = -1 \)[/tex].
Now, let's examine if there are any complex solutions by considering the cubic roots in the complex plane. The other roots come from the fact that the complex cube roots of 1 are:
[tex]\[ e^{2\pi i/3} \text{ and } e^{-2\pi i/3} \][/tex]
Thus, in a more general form, the solutions will be:
[tex]\[ x + 3 = 2 \][/tex]
[tex]\[ x + 3 = 2 e^{2\pi i/3} \][/tex]
[tex]\[ x + 3 = 2 e^{-2\pi i/3} \][/tex]
Solving these we get:
[tex]\[ x = -1 \][/tex]
[tex]\[ x = -3 + 2 e^{2\pi i/3} \][/tex]
[tex]\[ x = -3 + 2 e^{-2\pi i/3} \][/tex]
This results in the specific complex solutions:
[tex]\[ x = -4 - \sqrt{3}i \][/tex]
[tex]\[ x = -4 + \sqrt{3}i \][/tex]
Therefore, the algebraic solutions are:
[tex]\[ x = -1, -4 - \sqrt{3}i, -4 + \sqrt{3}i \][/tex]
### Graphical Solution
To graphically find the solution, we need to plot [tex]\( f(x) = (x+3)^3 + 4 \)[/tex] and find the points where it intersects the horizontal line [tex]\( y = 12 \)[/tex].
Here is a table that highlights some values of [tex]\( x \)[/tex] and their corresponding [tex]\( y \)[/tex] values:
| [tex]\( x \)[/tex] | [tex]\( y \)[/tex] |
|---------|-----------------|
| -11 | -508 |
| -10 | -339 |
| -9 | -212 |
| -8 | -121 |
| -7 | -60 |
| -6 | -23 |
| -5 | -4 |
| -4 | 3 |
| -3 | 4 |
| -2 | 5 |
| -1 | 12 |
| 0 | 31 |
| 1 | 68 |
| 2 | 129 |
| 3 | 220 |
| 4 | 347 |
| 5 | 516 |
| 6 | 733 |
| 7 | 1004 |
| 8 | 1335 |
From the table, it is evident that [tex]\( y = 12 \)[/tex] when [tex]\( x = -1 \)[/tex].
On a graph, we would draw the function [tex]\( f(x) \)[/tex] (which looks like a cubic function shifted left by 3 units and up 4 units) and the horizontal line [tex]\( y = 12 \)[/tex]. The intersection point at [tex]\( x = -1 \)[/tex] corresponds to [tex]\( y = 12 \)[/tex], confirming our algebraic solution.
### Conclusion
Algebraic solutions:
[tex]\[ x = -1, \][/tex]
[tex]\[ x = -4 - \sqrt{3}i, \][/tex]
[tex]\[ x = -4 + \sqrt{3}i. \][/tex]
Graphical solution:
From the table, we see that [tex]\( y = 12 \)[/tex] when [tex]\( x = -1 \)[/tex].
Hence, the real solution [tex]\( x = -1 \)[/tex] is confirmed by both methods.
