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Question 1209610: Suppose f is a function such that f(f(x)) = x for all x in the domain of f, and suppose that f(-3) = 0. If f(x) = ax + b, then find a and b. Find all solutions to f(x) = 5.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
1. **Use the given information about f(x):**
We know f(x) = ax + b and f(f(x)) = x. Substituting the expression for f(x) into the composite function:
f(f(x)) = f(ax + b) = a(ax + b) + b = a²x + ab + b = x
2. **Solve for a and b:**
For the equation a²x + ab + b = x to hold for all x, the coefficients of x must be equal, and the constant terms must be equal. This gives us the following system of equations:
* a² = 1
* ab + b = 0
From a² = 1, we have two possibilities: a = 1 or a = -1.
* **Case 1: a = 1**
Substituting into ab + b = 0, we get b + b = 0, so 2b = 0, and b = 0.
Thus, f(x) = x.
* **Case 2: a = -1**
Substituting into ab + b = 0, we get -b + b = 0. This equation is true for any value of b.
3. **Use f(-3) = 0 to find b (in the case a=-1):**
We are given that f(-3) = 0. Using f(x) = ax + b:
0 = a(-3) + b
If a = -1, then 0 = -(-3) + b, so 0 = 3 + b, and b = -3.
Thus, if a = -1, then b = -3, and f(x) = -x - 3.
4. **Determine which solution is correct:**
* **If f(x) = x:** Then f(-3) = -3, which contradicts the given information f(-3) = 0. So, f(x) = x is not a solution.
* **If f(x) = -x - 3:** Then f(-3) = -(-3) - 3 = 3 - 3 = 0, which matches the given information. Also, f(f(x)) = f(-x - 3) = -(-x - 3) - 3 = x + 3 - 3 = x. This solution works.
Therefore, a = -1 and b = -3, and f(x) = -x - 3.
5. **Solve f(x) = 5:**
We want to find x such that f(x) = 5. Using f(x) = -x - 3:
5 = -x - 3
8 = -x
x = -8
So, the solution to f(x) = 5 is x = -8.
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