Let `f:RrarrR` be defined by `f(x)=x^(2)+1`.
Find the pre-image of (i) 17 (ii) - 3.

Answer

To find the pre-image of the given values under the function \( f(x) = x^2 + 1 \), we will solve for \( x \) when \( f(x) \) is equal to the specified values.

### Step-by-Step Solution:

1. **Finding the pre-image of 17:**
- We set \( f(x) = 17 \).
- This gives us the equation:
\[ x^2 + 1 = 17 \]
- Subtract 1 from both sides:
\[ x^2 = 17 - 1 \]
\[ x^2 = 16 \]
- Now, take the square root of both sides:
\[ x = \pm \sqrt{16} \]
- Therefore, we have:
\[ x = \pm 4 \]
- Thus, the pre-images of 17 are \( x = 4 \) and \( x = -4 \).

2. **Finding the pre-image of -3:**
- We set \( f(x) = -3 \).
- This gives us the equation:
\[ x^2 + 1 = -3 \]
- Subtract 1 from both sides:
\[ x^2 = -3 - 1 \]
\[ x^2 = -4 \]
- Since the square of a real number cannot be negative, there are no real solutions for this equation. Therefore, there is no pre-image for -3.

### Summary of Results:
- The pre-images of 17 are \( x = 4 \) and \( x = -4 \).
- There is no pre-image for -3.