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# Algebra

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The function f(x) is defined for 1 \le x \le 5 as follows:

f(x) = 2x + 8 if 1 \le x \le 2

f(x) = 13 - 5x if 2 < x \le 3

f(x) = 20 - 14x if 3 < x \le 4

f(x) = 40 + 5x if 4 < x \le 5

Find all real numbers x such that f(x) = x.  If you find more than one answer, list them all, separated by commas.

Aug 14, 2024

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Ok, we can go one by one to solve these problems.

Hoever, let's note that if we have f(x) = x, then we set the right side of the function to equal x.

Then, we see if the x value satisfies the conditions given.

First, we have \(f(x) = 2x + 8\). Setting this to equal x, we find that \(x = 2x + 8\)

Solving for x, we find that \(x=-8\). This doesn't satisfy the condition \(1 \le x \le 2\), so it is not a valid solution.

Next, we have \(f(x) = 13 - 5x\). Setting this to equal x, we have \(x = 13 - 5x\)

Finding the value of x, we get \(x=13/6\). This satisfies the inerval, so it is a solution.

Third, we have \(f(x) = 20 - 14x\). Since this equals x, we can write \(x= 20 - 14x\)

Trying to find x, we have that \(x=20/15=4/3\). This does NOT satisfy the interval, so it is not a solution.

Lastly, we have \(f(x) = 40 + 5x\). setting this function to equal x, we have \(x = 40 + 5x\)

Solving for x, we get \(x=-10\), which is invalid.

Thus, the only value that works is 13/6.