If (f:\mathbb{R}\to\mathbb{R}) and (f(x)=x^2-2x), what is the value of (f(1+h)-f(1))?

Correct answer: (h^2)

The direct answer is A: \(h^2\). A function gives an output for each input. Put \(1+h\) into \(f(x)=x^2-2x\): \(f(1+h)=(1+h)^2-2(1+h)=1+2h+h^2-2-2h=h^2-1\). Also, \(f(1)=1^2-2(1)=-1\). Therefore, \(f(1+h)-f(1)=(h^2-1)-(-1)=h^2\). Option A is correct because it is exactly the simplified difference. Option B, \(h^2-1\), is only \(f(1+h)\), not the difference after subtracting \(f(1)\). Option C, \(2h+h^2\), does not result after the cancellation of the \(2h\) terms. Option D, \(h\), is missing the square term and is not obtained by the calculation. Remember: substitute both inputs, calculate both function values, and then subtract.