Answer :

By using the property of function, the result obtained is, When f(94) is divided by 1000, the remainder is 561

What is a function?

A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.

There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.

Here the functional equation is given by

[tex]f(x) + f(x - 1) = x^2[/tex]

Now,

[tex]f(x) + f(x - 1) = x^2[/tex]

[tex]f(x) = x^2 - f(x - 1)\\f(94) = 94^2 - f(93) = 94^2 - 93^2 + f(92) =\\94^2 - 93^2 + 92^2 - f(91) = 94^2 - 93^2 + 92^2 -91^2+f(90)[/tex]

[tex]94 + 93 + 92 + 91 + ..... +20^2 - f(19)[/tex]

[tex]94 + 93 + 92 + ..... + 21 + 400 - 94[/tex] [Since [tex]94 - 93 = 93 - 92 = .... = 1[/tex]]

[tex]\frac{74}{2}(21 + 94) + 400 -94[/tex]

[tex]37 \times 115 + 400 - 94\\4255 + 400 - 94\\4561[/tex]

So f(94) = 4561

When 4561 is divided by 1000, the remainder is 561

So, when f(94) is divided by 1000, the remainder is 561

To learn more about function, refer to the link:
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Complete Question

The function f has the property that, for each real number x

[tex]f(x)+f(x-1) = x^2[/tex]

If f(19)=94 what is the remainder when f(94) is divided by 1000?

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