Questions: Enter the inverse function if: y=1-3x and x ∈[-2 ; 2) Find the corresponding domain of the inverse function: Solve the task and fill in the required fields. Answer: Write the answer in the form of (7+6x) / 8 The inverse function is y= The corresponding domain of the inverse function is x ∈

Enter the inverse function if: y=1-3x and x ∈[-2 ; 2) Find the corresponding domain of the inverse function: Solve the task and fill in the required fields. Answer: Write the answer in the form of (7+6x) / 8 The inverse function is y= The corresponding domain of the inverse function is x ∈
Transcript text: Enter the inverse function if: $y=1-3 x$ and $x \in[-2 ; 2)$ Find the corresponding domain of the inverse function: Solve the task and fill in the required fields. Answer: Write the answer in the form of $(7+6 x) / 8$ The inverse function is $y=$ $\square$ 1 $\square$ The corresponding domain of the inverse function is $x \in$ $\square$ $\square$ $\square$ $\square$
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Solution

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Solution Steps

To find the inverse function of \( y = 1 - 3x \), we need to solve for \( x \) in terms of \( y \). Then, we will determine the domain of the inverse function based on the given domain of the original function.

Step 1: Find the Inverse Function

Given the function \( y = 1 - 3x \), we need to find its inverse. To do this, solve for \( x \) in terms of \( y \):

\[ y = 1 - 3x \]

Rearrange to solve for \( x \):

\[ y - 1 = -3x \implies x = \frac{1 - y}{3} \]

Thus, the inverse function is:

\[ f^{-1}(y) = \frac{1 - y}{3} \]

Step 2: Determine the Domain of the Inverse Function

The original function \( y = 1 - 3x \) has a domain \( x \in [-2, 2) \). To find the domain of the inverse function, we need to determine the range of the original function over this domain.

Calculate the range:

\[ \text{For } x = -2: \quad y = 1 - 3(-2) = 1 + 6 = 7 \] \[ \text{For } x = 2: \quad y = 1 - 3(2) = 1 - 6 = -5 \]

Thus, the range of the original function is \( y \in [-5, 7) \), which becomes the domain of the inverse function.

Final Answer

The inverse function is:

\[ f^{-1}(y) = \frac{1 - y}{3} \]

The corresponding domain of the inverse function is:

\[ x \in [-5, 7) \]

\[ \boxed{f^{-1}(y) = \frac{1 - y}{3}} \] \[ \boxed{x \in [-5, 7)} \]

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