Pregunta
Establish which of the following options is the inverse of \( f(x)=x^{3}+3 \) \( \begin{array}{l}f^{-1}(x)=\sqrt[3]{x-3} \\ f^{-1}(x)=\sqrt[3]{x}+3 \\ f^{-1}(x)=\sqrt[3]{x}-3 \\ f^{-1}(x)=\sqrt[3]{x+3}\end{array} \)
Ask by Rogers Riley. in the United States
Jan 23,2025
Solución de tutoría real
Respuesta verificada por el tutor
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The inverse function is \( f^{-1}(x) = \sqrt[3]{x - 3} \).
Solución
To find the inverse of the function \( f(x) = x^3 + 3 \), follow these steps:
1. **Start with the original function:**
\[
y = x^3 + 3
\]
2. **Solve for \( x \) in terms of \( y \):**
\[
y - 3 = x^3
\]
\[
x = \sqrt[3]{y - 3}
\]
3. **Express the inverse function:**
\[
f^{-1}(x) = \sqrt[3]{x - 3}
\]
Therefore, the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{x - 3}
\]
**Final Answer:**
\( f^{-1}(x)=\sqrt[3]{x-3} \)
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Beyond the Answer
To find the inverse of the function \( f(x) = x^3 + 3 \), we need to isolate \( x \) in terms of \( y \). 1. Start by replacing \( f(x) \) with \( y \): \[ y = x^3 + 3 \] 2. Next, rearrange the equation to solve for \( x \): \[ y - 3 = x^3 \] 3. Then, take the cube root of both sides: \[ x = \sqrt[3]{y - 3} \] 4. Finally, express the inverse function: \[ f^{-1}(x) = \sqrt[3]{x - 3} \] Thus, the correct inverse function is \( f^{-1}(x) = \sqrt[3]{x - 3} \). The answer is: \[ f^{-1}(x) = \sqrt[3]{x - 3} \]
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