The function f(x) is defined for as follows: . find all real numbers x such that f(x) = x. if you find more than one answer, list them all, separated by commas.

answer: 10/3, 5

explanation

essentially, we solve four times on each of the pieces' intervals.

for each piece, we find a value of x such that the piece will be equal to x. then, we see if that particular value of x is in the defined interval of the piece. if it is in the interval, then it is a valid answer. if it is not in the interval, then it is not a valid answer.

these are all linear pieces; therefore, each piece can have at most one x-value such that f(x) = x.

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for the first piece, , we solve

since this x value is not the interval , we disregard this solution.

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for the second piece, , we solve

since this x value is not in the interval , we disregard this solution.

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for the third piece, , we solve

since this x value is in the interval defined for this given piece, this is a solution to .

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for the fourth piece, , we solve

since this x-value is in the interval , this is a solution to the equation