Question

or
Complete the table of values for f(x)=4x2+6 and g(x)=2(3)x.
x f(x) g(x)
0
1
2
3
Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other?

Answer 1

Let's start by calculating the values for \( f(x) = 4x^2 + 6 \) and \( g(x) = 2(3^x) \).

### Calculating Values:
1. For \( x = 0 \):
\[
f(0) = 4(0)^2 + 6 = 6
\]
\[
g(0) = 2(3^0) = 2(1) = 2
\]

2. For \( x = 1 \):
\[
f(1) = 4(1)^2 + 6 = 4 + 6 = 10
\]
\[
g(1) = 2(3^1) = 2(3) = 6
\]

3. For \( x = 2 \):
\[
f(2) = 4(2)^2 + 6 = 4(4) + 6 = 16 + 6 = 22
\]
\[
g(2) = 2(3^2) = 2(9) = 18
\]

4. For \( x = 3 \):
\[
f(3) = 4(3)^2 + 6 = 4(9) + 6 = 36 + 6 = 42
\]
\[
g(3) = 2(3^3) = 2(27) = 54
\]

### Putting it all in the table:

\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
0 & 6 & 2 \\
1 & 10 & 6 \\
2 & 22 & 18 \\
3 & 42 & 54 \\
\hline
\end{array}
\]

### Conclusion:
- For \( x = 0 \), \( f(0) = 6 \) and \( g(0) = 2 \) (so \( f(0) > g(0) \)).
- For \( x = 1 \), \( f(1) = 10 \) and \( g(1) = 6 \) (so \( f(1) > g(1) \)).
- For \( x = 2 \), \( f(2) = 22 \) and \( g(2) = 18 \) (so \( f(2) > g(2) \)).
- For \( x = 3 \), \( f(3) = 42 \) and \( g(3) = 54 \) (so \( g(3) > f(3) \)).

As \( x \) continues to increase, \( g(x) \) (which has exponential growth) will eventually exceed \( f(x) \) (which has polynomial growth) since exponential functions grow faster than polynomial functions. Therefore, **g(x) will eventually exceed f(x)** as \( x \) gets larger.