To determine if the functions \( y = x^2 + 5 \) and \( y^2 = x - 5 \) are inverses of one another, we need to check if they satisfy the definition of inverse functions. Specifically, two functions \( f \) and \( g \) are inverses if \( f(g(x)) = x \) and \( g(f(x)) = x \).
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Start with the first function: \( y = x^2 + 5 \).
- We can express this as \( x = y - 5 \), leading to \( y = \sqrt{x - 5} \) (or \( y = -\sqrt{x - 5} \)) for the inverse, but here we will focus on the positive root since we are dealing with a square function.
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Now check the second function: \( y^2 = x - 5 \) implies \( y = \sqrt{x - 5} \) or \( y = -\sqrt{x - 5} \).
- Rearranging gives us \( x = y^2 + 5 \).
Next, we will check each function by substituting into each other:
1st check: \( f(g(x)) \)
Let \( g(x) = \sqrt{x - 5} \): \[ f(g(x)) = f(\sqrt{x - 5}) = (\sqrt{x - 5})^2 + 5 = (x - 5) + 5 = x. \]
2nd check: \( g(f(x)) \)
Let \( f(x) = x^2 + 5 \): \[ g(f(x)) = g(x^2 + 5) = \sqrt{(x^2 + 5) - 5} = \sqrt{x^2} = x \quad \text{(for } x \geq 0 \text{)}. \]
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, we conclude that the two functions are indeed inverses of each other.
Thus, the correct response is: Yes, they are inverses because all corresponding ordered pairs are reflections of one another across the line \( y = x \).
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