Question

complete the table of values for $f(x) = 4^x + 3$ and $g(x) = 6x + 7$.

$x$	$f(x)$	$g(x)$
2
3
4

both $f(x)$ and $g(x)$ grow as $x$ gets larger and larger. which function eventually exceeds the other?
$f(x) = 4^x + 3$
$g(x) = 6x + 7$
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Explanation:

Completing the Table of Values

First, we'll calculate the values of \( f(x) = 4^x + 3 \) and \( g(x) = 6x + 7 \) for \( x = 1, 2, 3, 4 \).

For \( x = 1 \):

• Calculating \( f(1) \):

Substitute \( x = 1 \) into \( f(x) \):
\( f(1) = 4^1 + 3 = 4 + 3 = 7 \)

• Calculating \( g(1) \):

Substitute \( x = 1 \) into \( g(x) \):
\( g(1) = 6(1) + 7 = 6 + 7 = 13 \)

For \( x = 2 \):

• Calculating \( f(2) \):

Substitute \( x = 2 \) into \( f(x) \):
\( f(2) = 4^2 + 3 = 16 + 3 = 19 \)

• Calculating \( g(2) \):

Substitute \( x = 2 \) into \( g(x) \):
\( g(2) = 6(2) + 7 = 12 + 7 = 19 \)

For \( x = 3 \):

• Calculating \( f(3) \):

Substitute \( x = 3 \) into \( f(x) \):
\( f(3) = 4^3 + 3 = 64 + 3 = 67 \)

• Calculating \( g(3) \):

Substitute \( x = 3 \) into \( g(x) \):
\( g(3) = 6(3) + 7 = 18 + 7 = 25 \)

For \( x = 4 \):

• Calculating \( f(4) \):

Substitute \( x = 4 \) into \( f(x) \):
\( f(4) = 4^4 + 3 = 256 + 3 = 259 \)

• Calculating \( g(4) \):

Substitute \( x = 4 \) into \( g(x) \):
\( g(4) = 6(4) + 7 = 24 + 7 = 31 \)

So the completed table is:

\( x \)	\( f(x) \)	\( g(x) \)
2	19	19
3	67	25
4	259	31

Determining Which Function Exceeds the Other

To determine which function eventually exceeds the other, we analyze the growth rates:

• \( f(x) = 4^x + 3 \) is an exponential function (base 4 > 1), so it grows exponentially.
• \( g(x) = 6x + 7 \) is a linear function (degree 1), so it grows linearly.

Exponential functions grow much faster than linear functions as \( x \) becomes large. From the table, we can see that for \( x = 3 \) and \( x = 4 \), \( f(x) \) is already larger than \( g(x) \) (and the gap will only increase as \( x \) gets larger).

So, the function \( f(x) = 4^x + 3 \) eventually exceeds \( g(x) = 6x + 7 \).

Final Answers

Completed Table:

\( x \)	\( f(x) \)	\( g(x) \)
2	19	19
3	67	25
4	259	31

Which function exceeds the other?

\( f(x) = 4^x + 3 \)

Answer:

Completing the Table of Values

First, we'll calculate the values of \( f(x) = 4^x + 3 \) and \( g(x) = 6x + 7 \) for \( x = 1, 2, 3, 4 \).

For \( x = 1 \):

• Calculating \( f(1) \):

Substitute \( x = 1 \) into \( f(x) \):
\( f(1) = 4^1 + 3 = 4 + 3 = 7 \)

• Calculating \( g(1) \):

Substitute \( x = 1 \) into \( g(x) \):
\( g(1) = 6(1) + 7 = 6 + 7 = 13 \)

For \( x = 2 \):

• Calculating \( f(2) \):

Substitute \( x = 2 \) into \( f(x) \):
\( f(2) = 4^2 + 3 = 16 + 3 = 19 \)

• Calculating \( g(2) \):

Substitute \( x = 2 \) into \( g(x) \):
\( g(2) = 6(2) + 7 = 12 + 7 = 19 \)

For \( x = 3 \):

• Calculating \( f(3) \):

Substitute \( x = 3 \) into \( f(x) \):
\( f(3) = 4^3 + 3 = 64 + 3 = 67 \)

• Calculating \( g(3) \):

Substitute \( x = 3 \) into \( g(x) \):
\( g(3) = 6(3) + 7 = 18 + 7 = 25 \)

For \( x = 4 \):

• Calculating \( f(4) \):

Substitute \( x = 4 \) into \( f(x) \):
\( f(4) = 4^4 + 3 = 256 + 3 = 259 \)

• Calculating \( g(4) \):

Substitute \( x = 4 \) into \( g(x) \):
\( g(4) = 6(4) + 7 = 24 + 7 = 31 \)

So the completed table is:

\( x \)	\( f(x) \)	\( g(x) \)
2	19	19
3	67	25
4	259	31

Determining Which Function Exceeds the Other

To determine which function eventually exceeds the other, we analyze the growth rates:

• \( f(x) = 4^x + 3 \) is an exponential function (base 4 > 1), so it grows exponentially.
• \( g(x) = 6x + 7 \) is a linear function (degree 1), so it grows linearly.

Exponential functions grow much faster than linear functions as \( x \) becomes large. From the table, we can see that for \( x = 3 \) and \( x = 4 \), \( f(x) \) is already larger than \( g(x) \) (and the gap will only increase as \( x \) gets larger).

So, the function \( f(x) = 4^x + 3 \) eventually exceeds \( g(x) = 6x + 7 \).

Final Answers

Completed Table:

\( x \)	\( f(x) \)	\( g(x) \)
2	19	19
3	67	25
4	259	31

Which function exceeds the other?

\( f(x) = 4^x + 3 \)