Let's solve each part of the problem step-by-step.
### Part (a): Solving [tex]\( f(x) = 12 \)[/tex]
The given function is:
[tex]\[ f(x) = (x + 3)^3 + 4 \][/tex]
We need to find [tex]\( x \)[/tex] when [tex]\( f(x) = 12 \)[/tex]; that is, we solve the equation:
[tex]\[ (x + 3)^3 + 4 = 12 \][/tex]
1. Step 1: Isolate the Cubic Term
[tex]\[ (x + 3)^3 + 4 = 12 \][/tex]
Subtract 4 from both sides:
[tex]\[ (x + 3)^3 = 8 \][/tex]
2. Step 2: Solve for [tex]\( x + 3 \)[/tex]
Take the cube root of both sides:
[tex]\[ x + 3 = \sqrt[3]{8} \][/tex]
Since the cube root of 8 is 2:
[tex]\[ x + 3 = 2 \][/tex]
3. Step 3: Solve for [tex]\( x \)[/tex]
Subtract 3 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x = 2 - 3 \][/tex]
[tex]\[ x = -1 \][/tex]
Thus, the solution is:
[tex]\[ x = -1 \][/tex]
This means that when [tex]\( f(x) = 12 \)[/tex], the value of [tex]\( x \)[/tex] is [tex]\( -1 \)[/tex].
### Filling in the Table
Based on the solution derived:
[tex]\[
\begin{tabular}{|l|l|}
\hline $x$ & $y$ \\
\hline -1 & 12 \\
\hline & \\
\hline & \\
\hline & \\
\hline & \\
\hline & \\
\hline & \\
\hline
\end{tabular}
\][/tex]
### Part (b): Solving [tex]\( g(x) = 12 \)[/tex]
The function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[ g(x) = f(x + 1) - 2 \][/tex]
We need to find [tex]\( x \)[/tex] when [tex]\( g(x) = 12 \)[/tex]; that is, we solve the equation:
[tex]\[ f(x + 1) - 2 = 12 \][/tex]
1. Step 1: Isolate [tex]\( f(x + 1) \)[/tex]
[tex]\[ f(x + 1) - 2 = 12 \][/tex]
Add 2 to both sides:
[tex]\[ f(x + 1) = 14 \][/tex]
2. Step 2: Use the Definition of [tex]\( f(x) \)[/tex]
Recall that [tex]\( f(x) = (x + 3)^3 + 4 \)[/tex]:
[tex]\[ f(x + 1) = ((x + 1) + 3)^3 + 4 \][/tex]
[tex]\[ f(x + 1) = (x + 4)^3 + 4 \][/tex]
So, we need to solve:
[tex]\[ (x + 4)^3 + 4 = 14 \][/tex]
3. Step 3: Isolate the Cubic Term
Subtract 4 from both sides:
[tex]\[ (x + 4)^3 = 10 \][/tex]
4. Step 4: Solve for [tex]\( x + 4 \)[/tex]
Take the cube root of both sides:
[tex]\[ x + 4 = \sqrt[3]{10} \][/tex]
5. Step 5: Solve for [tex]\( x \)[/tex]
Subtract 4 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt[3]{10} - 4 \][/tex]
This gives the numerical value:
[tex]\[ x \approx -1.8455653099681162 \][/tex]
Thus, when [tex]\( g(x) = 12 \)[/tex], the value of [tex]\( x \)[/tex] is approximately [tex]\( -1.8455653099681162 \)[/tex].
### Summary of the Solutions
- For [tex]\( f(x) = 12 \)[/tex], [tex]\( x = -1 \)[/tex].
- For [tex]\( g(x) = 12 \)[/tex], [tex]\( x \approx -1.8455653099681162 \)[/tex].
