Question - Determining the Nature of Functions

Given the function \( f(x) = x^2 - 5 \).

To determine if the function has real roots, we can find the discriminant.

The discriminant \( D \) is calculated as \( D = b^2 - 4ac \), where \( a = 1, b = 0, c = -5 \).

Thus, \( D = 0^2 - 4 \cdot 1 \cdot (-5) = 20 \).

Since \( D > 0 \), the function has two distinct real roots.

To find the roots, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \).

So, \( x = \frac{-0 \pm \sqrt{20}}{2 \cdot 1} = \frac{\pm 2\sqrt{5}}{2} = \pm \sqrt{5} \).

Therefore, the roots are \( x = \sqrt{5} \) and \( x = -\sqrt{5} \).