Let's proceed step by step to determine the month when the funds in the [tex]$f(x)$[/tex] bank account exceed those in the [tex]$g(x)$[/tex] bank account.
1. The functions provided for each of the accounts are:
- [tex]\( f(x) = 2^x \)[/tex]
- [tex]\( g(x) = 4x + 12 \)[/tex]
2. Evaluating each function at various months:
| Month (x) | [tex]\( f(x) = 2^x \)[/tex] | [tex]\( g(x) = 4x + 12 \)[/tex] |
|-----------|------------------|---------------------|
| 1 | [tex]\( 2^1 = 2 \)[/tex] | [tex]\( 4 \times 1 + 12 = 16 \)[/tex] |
| 2 | [tex]\( 2^2 = 4 \)[/tex] | [tex]\( 4 \times 2 + 12 = 20 \)[/tex] |
| 3 | [tex]\( 2^3 = 8 \)[/tex] | [tex]\( 4 \times 3 + 12 = 24 \)[/tex] |
| 4 | [tex]\( 2^4 = 16 \)[/tex] | [tex]\( 4 \times 4 + 12 = 28 \)[/tex] |
| 5 | [tex]\( 2^5 = 32 \)[/tex] | [tex]\( 4 \times 5 + 12 = 32 \)[/tex] |
| 6 | [tex]\( 2^6 = 64 \)[/tex] | [tex]\( 4 \times 6 + 12 = 36 \)[/tex] |
3. Compare the values of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] at each month:
- Month 1: [tex]\( f(1) = 2 \)[/tex] and [tex]\( g(1) = 16 \)[/tex] -> [tex]\( 2 < 16 \)[/tex]
- Month 2: [tex]\( f(2) = 4 \)[/tex] and [tex]\( g(2) = 20 \)[/tex] -> [tex]\( 4 < 20 \)[/tex]
- Month 3: [tex]\( f(3) = 8 \)[/tex] and [tex]\( g(3) = 24 \)[/tex] -> [tex]\( 8 < 24 \)[/tex]
- Month 4: [tex]\( f(4) = 16 \)[/tex] and [tex]\( g(4) = 28 \)[/tex] -> [tex]\( 16 < 28 \)[/tex]
- Month 5: [tex]\( f(5) = 32 \)[/tex] and [tex]\( g(5) = 32 \)[/tex] -> [tex]\( 32 = 32 \)[/tex]
- Month 6: [tex]\( f(6) = 64 \)[/tex] and [tex]\( g(6) = 36 \)[/tex] -> [tex]\( 64 > 36 \)[/tex]
4. We observe that in Month 6, the funds in the [tex]\( f(x) \)[/tex] bank account exceed those in the [tex]\( g(x) \)[/tex] bank account.
Therefore, the funds in the [tex]\( f(x) \)[/tex] bank account exceed those in the [tex]\( g(x) \)[/tex] bank account in Month 6.
