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.

 Jul 8, 2024

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 fuind 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 i 13/6. 

So our answer is 13/6. 


Thanks! :)

 Jul 8, 2024

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