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avatar+388 

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 11, 2024
 #1
avatar+759 
+1

We can go through every function to solve this problem. 

 

First, we have \(x=2x+8\), because we want f(x) to equal x. 

We solve this to find \(x=-8\). Unfortunately, this is not in the range given for x, so the solution is invalid. 

 

Now, we have \(x=13-5x\).

Solving for this, we have \(x=13/6\). This is in the range given, so it is a valid solution. 

 

Third, we have \(x=20-14x\)

We get \(x=4/3\). This is not in the range given, so the solution is invalid. 

 

Lastly, we have \(x=40+5x\)

Solving for this, we get \(x=-10\). This is not in the range, so it is invalid. 

 

So, the only solution is \(x=13/6\)

 

Thanks! :)

 Jun 11, 2024

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