Questions: Complete the table of values for f(x)=2(4)^x and g(x)=8x+6. x, f(x), g(x) 0, square, square 1, square, square 2, square, square 3, square, square Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other? f(x)=2(4)^x g(x)=8x+6

Complete the table of values for f(x)=2(4)^x and g(x)=8x+6.

x, f(x), g(x)
0, square, square
1, square, square
2, square, square
3, square, square

Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other?
f(x)=2(4)^x
g(x)=8x+6

Transcript text: Complete the table of values for $f(x)=2(4)^{x}$ and $g(x)=8 x+6$. \begin{tabular}{|c|l|l|} \hline $\boldsymbol{x}$ & $\boldsymbol{f ( x )}$ & $\boldsymbol{g ( x )}$ \\ \hline $\mathbf{0}$ & $\square$ & $\square$ \\ \hline 1 & $\square$ & $\square$ \\ \hline 2 & $\square$ & $\square$ \\ \hline 3 & $\square$ & $\square$ \\ \hline \end{tabular} Both $f(x)$ and $g(x)$ grow as $x$ gets larger and larger. Which function eventually exceeds the other? $f(x)=2(4)^{x}$ $g(x)=8 x+6$ Submit Work it out

Solution

Solution Steps

Step 1: Calculate $ f(x) $ and $ g(x) $ for $ x = 0 $

For $ x = 0 $:

$ f(0) = 2(4)^0 = 2 \times 1 = 2 $
$ g(0) = 8(0) + 6 = 0 + 6 = 6 $

Step 2: Calculate $ f(x) $ and $ g(x) $ for $ x = 1 $

For $ x = 1 $:

$ f(1) = 2(4)^1 = 2 \times 4 = 8 $
$ g(1) = 8(1) + 6 = 8 + 6 = 14 $

Step 3: Calculate $ f(x) $ and $ g(x) $ for $ x = 2 $

For $ x = 2 $:

$ f(2) = 2(4)^2 = 2 \times 16 = 32 $
$ g(2) = 8(2) + 6 = 16 + 6 = 22 $

Step 4: Calculate $ f(x) $ and $ g(x) $ for $ x = 3 $

For $ x = 3 $:

$ f(3) = 2(4)^3 = 2 \times 64 = 128 $
$ g(3) = 8(3) + 6 = 24 + 6 = 30 $

Step 5: Complete the Table

The completed table is as follows:

\[ \begin{tabular}{|c|l|l|} \hline $\boldsymbol{x}$ & $\boldsymbol{f(x)}$ & $\boldsymbol{g(x)}$ \\ \hline $\mathbf{0}$ & 2 & 6 \\ \hline 1 & 8 & 14 \\ \hline 2 & 32 & 22 \\ \hline 3 & 128 & 30 \\ \hline \end{tabular} \]

Step 6: Determine Which Function Eventually Exceeds the Other

As $ x $ increases, $ f(x) = 2(4)^x $ grows exponentially, while $ g(x) = 8x + 6 $ grows linearly. Exponential functions grow faster than linear functions, so $ f(x) $ will eventually exceed $ g(x) $.

Final Answer

The completed table is:

\[ \boxed{ \begin{tabular}{|c|l|l|} \hline $\boldsymbol{x}$ & $\boldsymbol{f(x)}$ & $\boldsymbol{g(x)}$ \\ \hline $\mathbf{0}$ & 2 & 6 \\ \hline 1 & 8 & 14 \\ \hline 2 & 32 & 22 \\ \hline 3 & 128 & 30 \\ \hline \end{tabular} } \]

The function $ f(x) = 2(4)^x $ eventually exceeds $ g(x) = 8x + 6 $.

\[ \boxed{f(x) \text{ eventually exceeds } g(x)} \]

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