`f:R->R` defined by `f(x) = x^2+5`

Options

A. `(x+5)^((1)/(3))`
B. `(x-5)^((1)/(3))`
C. `(5-x)^((1)/(3))`
D. `5-x`

Explanation

To find the inverse of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^3 + 5 \), we will follow these steps:

### Step 1: Set up the equation
Let \( y = f(x) \). Therefore, we have:
\[ y = x^3 + 5 \]

### Step 2: Rearrange the equation to solve for \( x \)
To find the inverse, we need to express \( x \) in terms of \( y \). First, we can rearrange the equation by isolating \( x^3 \):
\[ y - 5 = x^3 \]

### Step 3: Take the cube root of both sides
Next, we take the cube root of both sides to solve for \( x \):
\[ x = \sqrt[3]{y - 5} \]

### Step 4: Write the inverse function
Now that we have \( x \) in terms of \( y \), we can express the inverse function \( f^{-1}(x) \) by replacing \( y \) with \( x \):
\[ f^{-1}(x) = \sqrt[3]{x - 5} \]

### Conclusion
Thus, the inverse function is:
\[ f^{-1}(x) = \sqrt[3]{x - 5} \]

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