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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.

Apr 14, 2024

#1
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For this problem, we have to go throug every function and find when f(x) = x.

First off, we have \(f(x)=2x+8\)

We have to solve \(2x+8=x\), in which we get \(x=-8\). However, -8 isn't in the range \( 1 \le x \le 2\), so it isn't a valid solution.

Next, we have \(f(x) = 13 - 5x \)

Solving for \(13-5x=x\), we get \(x=13/6\). Indeed, 13/6 is in the range \( 2 < x \le 3\), so it is a valid solution.

After, we have \(f(x) = 20 - 14x\)

Solving for \(20-14x=x\), we get \(x=4/3\), which is not a valid solution since it's not in the range \(3 < x \le 4\)

Lastly, we have \(f(x)= 40 + 5x \)

We have \(40+5x=x\), in which we get \(x=-10\), which is not a valid solution.

So, we only have x = 13/6

Thanks! :)

Jun 2, 2024