Questions: Given g(x)=3x^3, a. Calculate g(3) and compare this value with g(6). g(3)= and g(6)=. b. Calculate g(4) and compare this value with g(8). g(4)= and g(8)=. c. What happens to the value of g(x) if x doubles in value? Enter your answer using an integer. The value of g(x) is . d. What happens to g(x) if x is divided by 2? Enter your answer using an integer. The value of g(x) is .

Given g(x)=3x^3,
a. Calculate g(3) and compare this value with g(6).
g(3)= and g(6)=.
b. Calculate g(4) and compare this value with g(8).
g(4)= and g(8)=.
c. What happens to the value of g(x) if x doubles in value? Enter your answer using an integer.

The value of g(x) is .
d. What happens to g(x) if x is divided by 2? Enter your answer using an integer.
The value of g(x) is .

Transcript text: Given $g(x)=3 x^{3}$, a. Calculate $g(3)$ and compare this value with $g(6)$. $g(3)=$ $\square$ and $g(6)=$ $\square$. b. Calculate $g(4)$ and compare this value with $g(8)$. $g(4)=$ $\square$ and $g(8)=$ $\square$. c. What happens to the value of $g(x)$ if $x$ doubles in value? Enter your answer using an integer. The value of $g(x)$ is $\square$. d. What happens to $g(x)$ if $x$ is divided by 2? Enter your answer using an integer. The value of $g(x)$ is $\square$.

Solution

Solution Steps

Solution Approach

a. To calculate $ g(3) $ and $ g(6) $, substitute $ x = 3 $ and $ x = 6 $ into the function $ g(x) = 3x^3 $ and compute the results. Compare the two values obtained.

b. Similarly, substitute $ x = 4 $ and $ x = 8 $ into the function $ g(x) = 3x^3 $ to find $ g(4) $ and $ g(8) $. Compare these values.

c. To determine what happens to $ g(x) $ when $ x $ doubles, consider the function $ g(2x) = 3(2x)^3 $ and compare it to $ g(x) = 3x^3 $. Calculate the ratio of $ g(2x) $ to $ g(x) $.

Step 1: Calculate $ g(3) $ and $ g(6) $

We have the function $ g(x) = 3x^3 $.

Calculating $ g(3) $: \[ g(3) = 3(3)^3 = 3 \cdot 27 = 81 \]

Calculating $ g(6) $: \[ g(6) = 3(6)^3 = 3 \cdot 216 = 648 \]

Step 2: Calculate $ g(4) $ and $ g(8) $

Now, we calculate $ g(4) $ and $ g(8) $.

Calculating $ g(4) $: \[ g(4) = 3(4)^3 = 3 \cdot 64 = 192 \]

Calculating $ g(8) $: \[ g(8) = 3(8)^3 = 3 \cdot 512 = 1536 \]

Step 3: Analyze the effect of doubling $ x $

To determine what happens to $ g(x) $ when $ x $ doubles, we compare $ g(2x) $ to $ g(x) $.

Calculating $ g(2x) $: \[ g(2x) = 3(2x)^3 = 3 \cdot 8x^3 = 24x^3 \]

The ratio of $ g(2x) $ to $ g(x) $ is: \[ \frac{g(2x)}{g(x)} = \frac{24x^3}{3x^3} = 8 \]