Question

Complete the table of values for f(x)=4(2)x and g(x)=5x+8.
x f(x) g(x)
1
2
3
4
Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other?
Scratchpad
Skip to next que

Answer 1

Let's first calculate the values of \( f(x) \) and \( g(x) \) for each \( x \) from 1 to 4.

1. For \( x = 1 \):
- \( f(1) = 4(2^1) = 4 \times 2 = 8 \)
- \( g(1) = 5(1) + 8 = 5 + 8 = 13 \)

2. For \( x = 2 \):
- \( f(2) = 4(2^2) = 4 \times 4 = 16 \)
- \( g(2) = 5(2) + 8 = 10 + 8 = 18 \)

3. For \( x = 3 \):
- \( f(3) = 4(2^3) = 4 \times 8 = 32 \)
- \( g(3) = 5(3) + 8 = 15 + 8 = 23 \)

4. For \( x = 4 \):
- \( f(4) = 4(2^4) = 4 \times 16 = 64 \)
- \( g(4) = 5(4) + 8 = 20 + 8 = 28 \)

Now we can complete the table of values:

\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
1 & 8 & 13 \\
2 & 16 & 18 \\
3 & 32 & 23 \\
4 & 64 & 28 \\
\hline
\end{array}
\]

### Observations:
- At \( x = 1 \): \( f(1) = 8 \), \( g(1) = 13 \) → \( g(x) > f(x) \)
- At \( x = 2 \): \( f(2) = 16 \), \( g(2) = 18 \) → \( g(x) > f(x) \)
- At \( x = 3 \): \( f(3) = 32 \), \( g(3) = 23 \) → \( f(x) > g(x) \)
- At \( x = 4 \): \( f(4) = 64 \), \( g(4) = 28 \) → \( f(x) > g(x) \)

### Conclusion:
As \( x \) gets larger, the function \( f(x) = 4(2^x) \) eventually exceeds \( g(x) = 5x + 8 \). The exponential growth of \( f(x) \) means that it will surpass the linear growth of \( g(x) \) after \( x = 2 \).