The function f(x)=1/2(1/3x−4)^3−1. Which of the following describes the transformations of the inverse function f−1(x) applied to its parent function?(1 point)

Question

The function f(x)=1/2(1/3x−4)^3−1. Which of the following describes the transformations of the inverse function f−1(x) applied to its parent function?(1 point)
a. horizontal translation 1 unit left, vertical translation 4 units up, vertical compression by a factor of 1/2, and horizontal compression by a factor of 1/2

b. horizontal translation 1 unit left, vertical translation 4 units up, vertical stretch by a factor of 3, and horizontal stretch by a factor of 2

c. horizontal translation 1 unit left, vertical translation 4 units down, vertical stretch by a factor of 3, and horizontal compression by a factor of 1212

d. horizontal translation 1 unit left, vertical translation 4 units up, vertical stretch by a factor of 3, and horizontal compression by a factor of 1/2

GPT-4o mini · Answer

To find the transformations of the inverse function \( f^{-1}(x) \) from the function given by

\[ f(x) = \frac{1}{2} \left(\frac{1}{3}x - 4\right)^3 - 1, \]

we will analyze the transformations step by step.

Step 1: Identify the parent function

The parent function here is a cubic function, which is \( g(x) = x^3 \).

Step 2: Identify transformations in \( f(x) \)

The term \( \left(\frac{1}{3}x - 4\right) \) indicates:
- A horizontal stretch by a factor of 3 (due to the factor of \( \frac{1}{3} \)) since we consider \( x \to 3x \) for the inverse.
- A horizontal translation of 4 units to the right (the negative sign indicates a leftward translation).
The \( \frac{1}{2} \) factor in front of the cubic indicates:
- A vertical compression by a factor of \( \frac{1}{2} \).
The \( -1 \) at the end indicates:
- A vertical translation 1 unit down.

Combining these transformations gives us for the function \( f(x) \):

Horizontal translation: 4 units to the right (which is 4 in the original, so -4 for inverse).
Vertical compression by \( \frac{1}{2} \).
Vertical translation: Down 1 unit.

Step 3: Inverse function transformations

When finding the transformations applied to \( f^{-1}(x) \), we consider the inverse operation effects on translations and stretches/compressions:

Horizontal translation:
- Since the original function \( f(x) \) has a horizontal translation of 4 units to the right, the inverse function will reflect this as a horizontal translation of 4 units to the left.
Vertical compression:
- This does not change as it will still remain a vertical compression by a factor of \( \frac{1}{2} \).
Vertical translation:
- Since the original function moved down by 1 unit, the inverse function will move up by 1 unit.

Step 4: Review options

To match the transformations on \( f^{-1}(x) \):

Based on above steps, we want a horizontal translation 4 units left, vertical translation 1 unit up, vertical compression by a factor of \( \frac{1}{2} \).

Now let's analyze the options: None of the options fit exactly but option (d) "horizontal translation 1 unit left, vertical translation 4 units up, vertical stretch by a factor of 3, and horizontal compression by a factor of \( \frac{1}{2} \)" aligns thematically, albeit with different specifics on translations.

Thus, the correct option considering the nature of transformations through inverse analysis would be: Option a. However be cautious, as the details should numerically match more accurately what gets distanced. None might match 100% based on transformations reflectively.

Final inference guides us toward Option A, but validating your transformations to ensure accuracy is key before final confirmation, as manipulations influence expected inversion.