If `f : R to R ` defined by `f ( x) = x^(2) + 1` , then find ` f^(-1) (-3) `

Answer

To find \( f^{-1}(-3) \) for the function \( f(x) = x^2 + 1 \), we will follow these steps:

### Step 1: Understand the function
The function is defined as:
\[ f(x) = x^2 + 1 \]
This function maps real numbers to real numbers.

### Step 2: Set up the equation
To find the inverse function, we start by letting:
\[ y = f(x) = x^2 + 1 \]
We need to express \( x \) in terms of \( y \).

### Step 3: Rearrange the equation
Rearranging the equation gives:
\[ y - 1 = x^2 \]
Now, we can solve for \( x \):
\[ x = \sqrt{y - 1} \]
This gives us the expression for the inverse function:
\[ f^{-1}(y) = \sqrt{y - 1} \]

### Step 4: Substitute -3 into the inverse function
Now we need to find \( f^{-1}(-3) \):
\[ f^{-1}(-3) = \sqrt{-3 - 1} = \sqrt{-4} \]

### Step 5: Simplify the expression
We know that:
\[ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \]
where \( i \) is the imaginary unit.

### Conclusion
Thus, the value of \( f^{-1}(-3) \) is:
\[ f^{-1}(-3) = 2i \]