If `f:RtoR` is a function defined by `f(x)=x^(3)+5` then `f^(-1)(x)` is

Options

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

Answer

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
We start by letting \( y = f(x) \). Therefore, we have:
\[ y = x^3 + 5 \]

### Step 2: Solve for \( x \)
To find the inverse function, we need to express \( x \) in terms of \( y \). Rearranging the equation gives us:
\[ x^3 = y - 5 \]

### Step 3: Take the cube root
Next, we take the cube root of both sides to solve for \( x \):
\[ x = (y - 5)^{1/3} \]

### Step 4: Write the inverse function
Since we have expressed \( x \) in terms of \( y \), we can write the inverse function \( f^{-1}(y) \):
\[ f^{-1}(y) = (y - 5)^{1/3} \]

### Step 5: Replace \( y \) with \( x \)
To express the inverse function in standard form, we replace \( y \) with \( x \):
\[ f^{-1}(x) = (x - 5)^{1/3} \]

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