Let `f: R->R` be a function given by `f(x)=x^2+1.` Find: `f^(-1){-5}`

Explanation

To solve the problem of finding \( f^{-1}(-5) \) for the function \( f(x) = x^2 + 1 \), we will follow these steps:

### Step 1: Understand the function
The function \( f(x) = x^2 + 1 \) is defined for all real numbers \( x \). The output of this function is always \( x^2 + 1 \), which means the minimum value of \( f(x) \) occurs when \( x = 0 \), giving \( f(0) = 0^2 + 1 = 1 \). Therefore, the range of \( f \) is \( [1, \infty) \).

### Step 2: Determine if -5 is in the range of f
Since the range of \( f(x) \) is \( [1, \infty) \), we can see that -5 is not in this range. This means there is no real number \( x \) such that \( f(x) = -5 \).

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