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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.

Jun 2, 2024

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We basically just have to go thorugh every function and find when f(x) = x.

First, we have f(x) = 2x + 8.

We basically have \(2x+8=x\). Solving for this, we get \(x=-8\). Unfortunately, that is not in the range \(1 \le x \le 2\), so that wouldn't work.

Second, we have f(x) = 13 - 5x.

We have \(13-5x=x\). Solving this, we get \(x=13/6\). This is in the range \(2 < x \le 3\), so 13/6 is a valid solution

Third, we have f(x) = 20 - 14x.

We have \(20-14x=x\). Solving this, we get \(x=4/3\). This is not in the range of \(3 < x \le 4\), so it is not a valid solution.

Lastly, we have f(x) = 40 + 5x.

We have \(40+5x=x\). Solving this, we get \(x = -10\). This is not in the range \( 4 < x \le 5\), so it is not a valid solution.

So finally, we have \(x=13/6\)

Thanks! :)

Jun 2, 2024
edited by NotThatSmart  Jun 2, 2024