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.
To find the values of x such that f(x) = x, we can substitute each function definition of f(x) into the equation f(x) = x and solve for x.
1. For x in the range 1 ≤ x ≤ 2:
2x + 8 = x
Subtracting x from both sides: 2x - x + 8 = 0
Simplifying: x + 8 = 0
Subtracting 8 from both sides: x = -8
2. For x in the range 2 < x ≤ 3:
13 - 5x = x
Adding 5x to both sides: 13 = 6x
Dividing both sides by 6: x = 13/6
3. For x in the range 3 < x ≤ 4:
20 - 14x = x
Adding 14x to both sides: 20 = 15x
Dividing both sides by 15: x = 20/15 = 4/3
4. For x in the range 4 < x ≤ 5:
40 + 5x = x
Subtracting 5x from both sides: 40 = -4x
Dividing both sides by -4: x = -10
Therefore, the values of x for which f(x) = x are: -8, 13/6, 4/3, -10.