Grades 11-12 Video Solutions 2026-2026_11-12

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Question 26. A function f has the property that for every real number x, f of x plus 10 equals f of x, and f of 6 minus x equals negative f of x. The value of f of 27 is 9. What is the value of f of 9 plus f of 13? Is it a, negative 27, b, negative 9, c, negative 3, d, 3, or e, 9? Rule number one of our function, f of x plus 10 equals f of x, tells us that two inputs with a difference of 10 will have the same output. We can apply this rule several times to f of 27, subtracting 10 from x each time to get that f of negative 3 equals 9. Then we can apply the second rule, f of 6 minus x equals negative f of x, with x equals negative 3, to get that f of 6 minus negative 3 equals negative f of negative 3. Simplifying this gives us f of 9 equals negative 9. Next, notice that for the second rule, if we use positive 3 for x, we get f of 6 minus 3 equals negative f of 3. Simplifying this gives us f of 3 equals negative f of 3. The only value that satisfies this would be 0, so f of 3 is equal to 0. Then we can apply rule one and add 10 to get that f of 13 is 0. Finally, we can add f of 9 and f of 13 together to get negative 9, which is answer choice b.

Keywords

periodic function

functional equation

symmetry

f(9)

f(13)