Determine if the cubic function and the cube root function are inverses.

Question

(1 point)
Responses

Yes, all corresponding ordered pairs are reflections of one another across the line y=x
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Yes, all corresponding ordered pairs are reflections of one another across the line y equals x .

No, not all corresponding ordered pairs are reflections of one another across the line y=x
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=
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.
No, not all corresponding ordered pairs are reflections of one another across the line y equals x .

No, not all corresponding ordered pairs are reflections of one another across the line y=−x
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=
−
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No, not all corresponding ordered pairs are reflections of one another across the line y equals negative x .

Yes, all corresponding ordered pairs are reflections of one another across the line y=−x
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=
−
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Yes, all corresponding ordered pairs are reflections of one another across the line y equals negative x .

Answer 1

To determine if the cubic function \( y = x^3 \) and the cube root function \( y = \sqrt[3]{x} \) are inverses, we need to check if the composition of the two functions yields the identity function.

1. First check: Composition of \( f \) and \( g \)
   • Let \( f(x) = x^3 \) (the cubic function) and \( g(x) = \sqrt[3]{x} \) (the cube root function).
   • Compute \( g(f(x)) = g(x^3) = \sqrt[3]{x^3} = x \).
   • Now compute \( f(g(x)) = f(\sqrt[3]{x}) = (\sqrt[3]{x})^3 = x \).

Since both compositions yield the identity function \( x \), we confirm that the cubic function and the cube root function are indeed inverses of each other.

Thus, the correct response is:

Yes, all corresponding ordered pairs are reflections of one another across the line \( y = x \).