Determine if y=x2+5

Question

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
.

Answer 1

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.