Asked by hudson
Determine if y=x2+5
and y2=x−5
are inverses of each other.
(1 point)
Responses
No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y=−x
.
No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y equals negative x .
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y=x
.
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y equals x .
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y=−x
.
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y equals negative x .
No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y=x
.
and y2=x−5
are inverses of each other.
(1 point)
Responses
No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y=−x
.
No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y equals negative x .
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y=x
.
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y equals x .
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y=−x
.
Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line y equals negative x .
No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y=x
.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine if \( y = x^2 + 5 \) and \( y^2 = x - 5 \) are inverses of each other, we can analyze the relationships between these equations.
1. **Rewrite the first equation in terms of \( x \)**:
\[
y = x^2 + 5 \implies x^2 = y - 5 \implies x = \sqrt{y - 5} \text{ (considering only the principal square root for simplicity)}
\]
2. **Rewrite the second equation in terms of \( y \)**:
\[
y^2 = x - 5 \implies x = y^2 + 5
\]
3. **Check the conditions for inverses**:
For functions to be inverses, the composition \( f(g(x)) = x \) and \( g(f(x)) = x \) must hold true.
- Define \( f(x) = x^2 + 5 \) and \( g(y) = y^2 + 5 \).
- Then:
\[
f(g(y)) = f(y^2 + 5) = (y^2 + 5)^2 + 5
\]
This does not simplify to \( y \).
Similarly, analyzing \( g(f(x)) \):
\[
g(f(x)) = g(x^2 + 5) = (x^2 + 5)^2 + 5
\]
This also does not simplify to \( x \).
In conclusion, the functions \( y = x^2 + 5 \) and \( y^2 = x - 5 \) are not inverses of each other because their respective compositions do not yield the original input.
### Therefore, the correct response is:
**No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y=x.**
1. **Rewrite the first equation in terms of \( x \)**:
\[
y = x^2 + 5 \implies x^2 = y - 5 \implies x = \sqrt{y - 5} \text{ (considering only the principal square root for simplicity)}
\]
2. **Rewrite the second equation in terms of \( y \)**:
\[
y^2 = x - 5 \implies x = y^2 + 5
\]
3. **Check the conditions for inverses**:
For functions to be inverses, the composition \( f(g(x)) = x \) and \( g(f(x)) = x \) must hold true.
- Define \( f(x) = x^2 + 5 \) and \( g(y) = y^2 + 5 \).
- Then:
\[
f(g(y)) = f(y^2 + 5) = (y^2 + 5)^2 + 5
\]
This does not simplify to \( y \).
Similarly, analyzing \( g(f(x)) \):
\[
g(f(x)) = g(x^2 + 5) = (x^2 + 5)^2 + 5
\]
This also does not simplify to \( x \).
In conclusion, the functions \( y = x^2 + 5 \) and \( y^2 = x - 5 \) are not inverses of each other because their respective compositions do not yield the original input.
### Therefore, the correct response is:
**No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y=x.**
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