Let's first complete the missing values in the table for \( f(x) = 3x + 3 \) and \( g(x) = 0.5(2)^x \).
Completing the Table
For \( f(x) \):
- When \( x = 6 \): \[ f(6) = 3(6) + 3 = 18 + 3 = 21 \]
For \( g(x) \):
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When \( x = 6 \): \[ g(6) = 0.5(2)^6 = 0.5 \times 64 = 32 \]
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When \( x = 7 \): \[ g(7) = 0.5(2)^7 = 0.5 \times 128 = 64 \]
Now, updating the table:
| x | f(x) = 3x + 3 | g(x) = 0.5(2)^x | |---|----------------|------------------| | 0 | 3 | 0.5 | | 1 | 6 | 1 | | 2 | 9 | 2 | | 3 | 12 | 4 | | 4 | 15 | 8 | | 5 | 18 | 16 | | 6 | 21 | 32 | | 7 | 24 | 64 |
Analyzing Function Values
Now we determine where the functions intersect.
- The two functions are equal when \( 3x + 3 = 0.5(2)^x \).
- We can check the values in the table:
- At \( x = 0 \): \( f(0) = 3 \), \( g(0) = 0.5 \) → \( f(0) > g(0) \)
- At \( x = 1 \): \( f(1) = 6 \), \( g(1) = 1 \) → \( f(1) > g(1) \)
- At \( x = 2 \): \( f(2) = 9 \), \( g(2) = 2 \) → \( f(2) > g(2) \)
- At \( x = 3 \): \( f(3) = 12 \), \( g(3) = 4 \) → \( f(3) > g(3) \)
- At \( x = 4 \): \( f(4) = 15 \), \( g(4) = 8 \) → \( f(4) > g(4) \)
- At \( x = 5 \): \( f(5) = 18 \), \( g(5) = 16 \) → \( f(5) > g(5) \)
- At \( x = 6 \): \( f(6) = 21 \), \( g(6) = 32 \) → \( g(6) > f(6) \)
From this analysis, we can see the functions intersect between \( x = 5 \) and \( x = 6 \).
Completing Sentences
Now we can fill in the sentences:
Near \( x = 0 \), f(x) is the function with greater values. Then, for some x-value between 5 and 6, the graphs intersect. Beyond that point of intersection, g(x) has greater values.