3. Let k, m, n ∈ N such that k < n. Use a shortest-paths argument to find a formula for the number of increasing functions from f : [m] → [n] such that f(x) < x + k for all x. Your answer should have the form (a choose b) - (c choose d)

step-by-step explanation:

to determine whether a given function is even, we simply substitute x with -x in the expression. if the result after simplification is identical to the original equation, then the function is said to be even other wise odd.

f(-x) = (-x)^2 - (-x) + 8

      = x^2 + x +8

the resulting function is not identical to the original one;

f(-x) ≠ f(x)

therefore, f(x) is not an even function

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