Question

complete the table of values for $f(x) = 3x + 5$ and $g(x) = 3(2)^x$.

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

both $f(x)$ and $g(x)$ grow as $x$ gets larger and larger. which function eventually exceeds the other?

• $f(x) = 3x + 5$
• $g(x) = 3(2)^x$

Explanation:

Part 1: Completing the Table of Values

For \( f(x) = 3x + 5 \):

Step 1: When \( x = 1 \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1) = 3(1) + 5 = 3 + 5 = 8 \)

Step 2: When \( x = 2 \)

Substitute \( x = 2 \) into \( f(x) \):
\( f(2) = 3(2) + 5 = 6 + 5 = 11 \)

Step 3: When \( x = 3 \)

Substitute \( x = 3 \) into \( f(x) \):
\( f(3) = 3(3) + 5 = 9 + 5 = 14 \)

Step 4: When \( x = 4 \)

Substitute \( x = 4 \) into \( f(x) \):
\( f(4) = 3(4) + 5 = 12 + 5 = 17 \)

For \( g(x) = 3(2)^x \):

Step 1: When \( x = 1 \)

Substitute \( x = 1 \) into \( g(x) \):
\( g(1) = 3(2)^1 = 3 \times 2 = 6 \)

Step 2: When \( x = 2 \)

Substitute \( x = 2 \) into \( g(x) \):
\( g(2) = 3(2)^2 = 3 \times 4 = 12 \)

Step 3: When \( x = 3 \)

Substitute \( x = 3 \) into \( g(x) \):
\( g(3) = 3(2)^3 = 3 \times 8 = 24 \)

Step 4: When \( x = 4 \)

Substitute \( x = 4 \) into \( g(x) \):
\( g(4) = 3(2)^4 = 3 \times 16 = 48 \)

The completed table is:

\( x \)	\( f(x) \)	\( g(x) \)
2	11	12
3	14	24
4	17	48

Part 2: Determining Which Function Exceeds the Other

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

• \( f(x) = 3x + 5 \) is a linear function (degree 1 polynomial), so it grows at a constant rate (slope of 3).
• \( g(x) = 3(2)^x \) is an exponential function (base 2 > 1), so it grows at an increasing rate (the rate of growth itself increases as \( x \) increases).

From the table:

• At \( x = 1 \), \( f(1) = 8 \) and \( g(1) = 6 \) ( \( f(x) \) is larger).
• At \( x = 2 \), \( f(2) = 11 \) and \( g(2) = 12 \) ( \( g(x) \) becomes larger).
• For \( x = 3, 4 \), \( g(x) \) is significantly larger than \( f(x) \).

Since exponential functions grow faster than linear functions for large values of \( x \), \( g(x) = 3(2)^x \) will eventually exceed \( f(x) = 3x + 5 \) as \( x \) gets larger and larger.

Answer:

(Table Completion):

\( x \)	\( f(x) \)	\( g(x) \)
2	11	12
3	14	24
4	17	48