Question
The quadratic function y=2x2 is reflected across the x-axis to form the function y=−2x2 . Which response best describes the effect this would have on the inverse of the function?(1 point) Responses The inverse function would be rotated by −90 degrees. The inverse function would be rotated by negative 90 degrees. The inverse function would be rotated by 90 degrees. The inverse function would be rotated by 90 degrees. The inverse would be reflected across the y-axis. The inverse would be reflected across the y -axis. The inverse would be reflected across the x-axis. The inverse would be reflected across the x -axis.
Answers
Answered by
GPT-4o mini
To find the effect of reflecting the function \( y = 2x^2 \) across the x-axis to form \( y = -2x^2 \) on its inverse, we start by considering the inverse of the original function.
The inverse of \( y = 2x^2 \) can be found by solving for \( x \):
1. Replace \( y \) with \( x \):
\[
x = 2y^2
\]
2. Solve for \( y \):
\[
y^2 = \frac{x}{2}
\]
\[
y = \pm \sqrt{\frac{x}{2}}
\]
For the function \( y = 2x^2 \), we typically restrict to the positive branch, so the inverse is:
\[
y = \sqrt{\frac{x}{2}}
\]
Now, after reflecting \( y = 2x^2 \) to get \( y = -2x^2 \), we consider the inverse of the new function \( y = -2x^2 \).
1. Replace \( y \) with \( x \):
\[
x = -2y^2
\]
2. Solve for \( y \):
\[
-2y^2 = x
\]
\[
y^2 = -\frac{x}{2}
\]
Since we can't take the square root of a negative number in the real number system, the result means that the inverse function for \( y = -2x^2 \) cannot exist in the same way as the original function without restricting the domain.
When the original function is reflected across the x-axis, the effect on any potential inverse would result in undefined behavior in the context of real numbers as it reflects into imaginary values or remains outside the range of its previous inverse definition.
To summarize, once you reflect the function across the x-axis, the behavior of the inverse will correspond to the reflected position but the nature of the function changes. Thus, the correct description would be that:
- The inverse function is now nonexistent or undefined within the real number system, but if forced to interpret the reflection, it would hint that it reflects across the x-axis.
However, none of the options you provided are perfectly suitable as an answer. The closest conceptually might be referring to the reflection across the x-axis but is not equivalent to a simple inverse operation in usual real terms. The inverse function wouldn't truly be reflected or defined anymore.
If you were to select based on the options, the reflection across the x-axis would best suit the transformation conceptually. Therefore, the response would be:
**The inverse would be reflected across the x-axis.**
The inverse of \( y = 2x^2 \) can be found by solving for \( x \):
1. Replace \( y \) with \( x \):
\[
x = 2y^2
\]
2. Solve for \( y \):
\[
y^2 = \frac{x}{2}
\]
\[
y = \pm \sqrt{\frac{x}{2}}
\]
For the function \( y = 2x^2 \), we typically restrict to the positive branch, so the inverse is:
\[
y = \sqrt{\frac{x}{2}}
\]
Now, after reflecting \( y = 2x^2 \) to get \( y = -2x^2 \), we consider the inverse of the new function \( y = -2x^2 \).
1. Replace \( y \) with \( x \):
\[
x = -2y^2
\]
2. Solve for \( y \):
\[
-2y^2 = x
\]
\[
y^2 = -\frac{x}{2}
\]
Since we can't take the square root of a negative number in the real number system, the result means that the inverse function for \( y = -2x^2 \) cannot exist in the same way as the original function without restricting the domain.
When the original function is reflected across the x-axis, the effect on any potential inverse would result in undefined behavior in the context of real numbers as it reflects into imaginary values or remains outside the range of its previous inverse definition.
To summarize, once you reflect the function across the x-axis, the behavior of the inverse will correspond to the reflected position but the nature of the function changes. Thus, the correct description would be that:
- The inverse function is now nonexistent or undefined within the real number system, but if forced to interpret the reflection, it would hint that it reflects across the x-axis.
However, none of the options you provided are perfectly suitable as an answer. The closest conceptually might be referring to the reflection across the x-axis but is not equivalent to a simple inverse operation in usual real terms. The inverse function wouldn't truly be reflected or defined anymore.
If you were to select based on the options, the reflection across the x-axis would best suit the transformation conceptually. Therefore, the response would be:
**The inverse would be reflected across the x-axis.**