Question

This table shows some values of the functions f(x) and g(x). Complete 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
7
Now, complete the sentences about the graphs of the two functions.
Near x=0,
is the function with greater values. Then, for some x-value between
, the graphs intersect. Beyond that point of intersection,
has greater values.

Answer 1

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 \( g(x) \):**
- When \( x = 6 \):
\[
g(6) = 0.5(2)^6 = 0.5 \times 64 = 32
\]

- 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.