If (f(x)=x^2-8x+20) and (g(x)=5), what is the minimum value of ((f+g)(x))?

Correct answer: (9)

The direct answer is option A, 9. Add the functions pointwise: \((f+g)(x)=f(x)+g(x)=x^2-8x+20+5=x^2-8x+25\). Complete the square: \(x^2-8x+25=(x-4)^2+9\). For every real number \(x\), a square such as \((x-4)^2\) is at least zero. Its smallest value is zero when \(x=4\). Thus the smallest possible value of the whole expression is \(0+9=9\). Option A is correct. Option B, 5, forgets the quadratic part and is not the minimum of the complete expression. Option C, 25, is the constant term before completing the square and is not the minimum; for example, at \(x=4\), the value is 9. Option D, 4, is the value of \(x\) where the minimum occurs, not the minimum value itself. The minimum is attained at \(x=4\), but the requested answer is the output value, 9. Memory cue: rewrite a quadratic as \((x-h)^2+k\); its minimum is \(k\) when the squared term is zero.