Discussion
\(\Delta\!\!\!|\,\) is a divided difference operator and \(\Delta\) is a forward difference operator, f(x) and g(x) are two functions and a and b are constants, then :
Answer: A
The correct property is linearity of the divided difference: the operator applied to \(af(x) + bg(x)\) equals a times its value on f plus b times its value on g.
A first divided difference is
\[ [x_{0}, x_{1}]f = \frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}} \]
and replacing f by \(af + bg\) splits the numerator into a times the f part plus b times the g part. Higher divided differences are built recursively from the same quotients, so they are linear too.
Option C mixes in the forward difference \(\Delta\), a different operator. Options B and D replace f and g by their product, which has nothing to do with a linear combination.
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