Questions: Write a formula for the function, (g(x)), described as follows: Use the function, (f(x)=x^3). Move the function to the right by 2 units and down by 3 units.

Write a formula for the function, (g(x)), described as follows: Use the function, (f(x)=x^3). Move the function to the right by 2 units and down by 3 units.

Transcript text: Write a formula for the function, $g(x)$, described as follows: Use the function, $f(x)=x^{3}$. Move the function to the right by 2 units and down by 3 units.

Solution

Solution Steps

To transform the function $ f(x) = x^3 $ by moving it to the right by 2 units, we replace $ x $ with $ (x - 2) $. To move the function down by 3 units, we subtract 3 from the entire function. Therefore, the transformed function $ g(x) $ is given by substituting these transformations into $ f(x) $.

Step 1: Define the Original Function

The original function is given by \[ f(x) = x^3. \]

Step 2: Apply the Horizontal Shift

To move the function to the right by 2 units, we replace $ x $ with $ (x - 2) $: \[ f(x - 2) = (x - 2)^3. \]

Step 3: Apply the Vertical Shift

Next, to move the function down by 3 units, we subtract 3 from the entire function: \[ g(x) = (x - 2)^3 - 3. \]

Step 4: Evaluate the Function at $ x = 5 $

Now, we evaluate $ g(5) $: \[ g(5) = (5 - 2)^3 - 3 = 3^3 - 3 = 27 - 3 = 24. \]