To determine if the cubic function \( f(x) = x^3 \) and the cube root function \( g(x) = \sqrt[3]{x} \) are inverses of each other, we can check if applying one function to the other returns the original input.
- Function Composition: We need to check if \( f(g(x)) = x \) and if \( g(f(x)) = x \).
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First, let's calculate \( f(g(x)) \): \[ f(g(x)) = f\left(\sqrt[3]{x}\right) = \left(\sqrt[3]{x}\right)^3 = x \]
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Now, let's calculate \( g(f(x)) \): \[ g(f(x)) = g(x^3) = \sqrt[3]{x^3} = x \]
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Since both compositions return the original input \( x \), we can conclude that \( f(x) = x^3 \) and \( g(x) = \sqrt[3]{x} \) are indeed inverse functions of each other.
- Reflection Across the Line \( y = x \): Since \( f \) and \( g \) are inverses, their corresponding ordered pairs will reflect across the line \( y = x \).
Thus, the correct response is:
Yes, all corresponding ordered pairs are reflections of one another across the line \( y = x \).