Let `f:R rarr R` be defined by `f(x)=x^(2)+1`, find the pre image of `17` and `- 3`, respectively, are

Answer

To find the pre-image of the values 17 and -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 means for any real number \( x \), \( f(x) \) will yield a value that is at least 1, since \( x^2 \) is always non-negative.

### Step 2: Find the Pre-image of 17
We need to find \( x \) such that:
\[ f(x) = 17 \]
This translates to:
\[ x^2 + 1 = 17 \]
Subtracting 1 from both sides gives:
\[ x^2 = 16 \]
Taking the square root of both sides results in:
\[ x = \pm 4 \]
Thus, the pre-images of 17 are:
\[ x = 4 \quad \text{and} \quad x = -4 \]

### Step 3: Find the Pre-image of -3
Next, we need to find \( x \) such that:
\[ f(x) = -3 \]
This translates to:
\[ x^2 + 1 = -3 \]
Subtracting 1 from both sides gives:
\[ x^2 = -4 \]
Since the square of a real number cannot be negative, there are no real solutions for this equation. Therefore, the pre-image of -3 does not exist.

### Summary of Results
- The pre-image of 17 is \( \{4, -4\} \).
- The pre-image of -3 does not exist.