Question

Determine if  y=x2+5  and y2=x−5  are inverses of each other.(1 point)ResponsesYes, 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.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 .No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line y=x.

Answer 1

To determine whether the functions \( y = x^2 + 5 \) and \( y^2 = x - 5 \) are inverses of each other, we can start by testing if they satisfy the condition for inverse functions: If \( f(g(x)) = x \) and \( g(f(x)) = x \), then \( f \) and \( g \) are inverses.

1. **Identify the Functions:**
- Let \( f(x) = x^2 + 5 \)
- Let \( g(y) = y^2 + 5 \) rearranged from \( y^2 = x - 5 \) to be \( x = y^2 + 5 \).

2. **Testing \( f(g(x)) \):**
\[
g(x) = x - 5 \implies f(g(x)) = f(x - 5) = (x - 5)^2 + 5
\]
Expanding:
\[
f(g(x)) = (x^2 - 10x + 25) + 5 = x^2 - 10x + 30
\]
Therefore, \( f(g(x)) \neq x \).

3. **Testing \( g(f(x)) \):**
\[
f(x) = x^2 + 5 \implies g(f(x)) = g(x^2 + 5) = (x^2 + 5)^2 - 5
\]
Expanding:
\[
g(f(x)) = (x^4 + 10x^2 + 25) - 5 = x^4 + 10x^2 + 20
\]
Therefore, \( g(f(x)) \neq x \).

Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \), the functions \( y = x^2 + 5 \) and \( y^2 = x - 5 \) are **not inverses of each other**.

The best response to the question is:
**No, they are not inverses because all corresponding ordered pairs are reflections of one another across the line \( y = -x \).** (The request is slightly misleading but clarifying that pairs are not reflecting any line correctly indicates they are not inverses.)