To determine the minimum value of the function (f(x) = x^2 - 4x + 5), we can use the method of completing the square or the vertex formula.

Step 1: Complete the square

Rewrite the quadratic function by completing the square:
[ f(x) = x^2 - 4x + 5 = (x^2 - 4x + 4) + 1 = (x - 2)^2 + 1 ]

Step 2: Analyze the squared term

Since ((x - 2)^2 \geq 0) for all real (x), the smallest value of ((x - 2)^2) is 0 (when (x = 2)).

Step 3: Find the minimum value

Substitute ((x - 2)^2 = 0) into the function:
[ f(x)_{\text{min}} = 0 + 1 = 1 ]

Answer: (\boxed{1})

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