Okay, so I need to find the value of b in the function q(x) = bx + 1, given that when I plug q(1) into f(x), the result is -3. Let me break this down step by step.

First, let's recall what the functions f(x) and q(x) are. The function f(x) is given as x - 3, which is a linear function. The function q(x) is another linear function, q(x) = bx + 1. The variable here is x, and b is the coefficient we need to find.

The problem states that f(q(1)) = -3. So, this means that if we first evaluate q at x = 1, then plug that result into f, the final output should be -3. So, the composition f(q(1)) equals -3. My goal is to find the value of b that makes this equation true.

Let me start by computing q(1). Since q(x) = bx + 1, substituting x = 1 gives q(1) = b*1 + 1 = b + 1. So, q(1) simplifies to b + 1. 

Now, this result, which is b + 1, is then the input to the function f(x). The function f(x) is x - 3, so substituting x with (b + 1) gives f(q(1)) = f(b + 1) = (b + 1) - 3. 

Simplifying that, (b + 1) - 3 is equal to b + 1 - 3, which further simplifies to b - 2. 

According to the problem, this composition equals -3. Therefore, we can set up the equation:

b - 2 = -3

Now, solving for b, we add 2 to both sides of the equation:

b - 2 + 2 = -3 + 2

Which simplifies to:

b = -1

So, the value of b should be -1. Let me check this to make sure I didn't make any mistakes.

First, compute q(1) with b = -1:

q(1) = (-1)(1) + 1 = -1 + 1 = 0

Then, compute f(q(1)) = f(0) = 0 - 3 = -3

Which matches the given condition. So, the calculation seems correct.

Wait, let me verify once again. If b is -1, then q(x) is -x + 1. Then q(1) is -1 + 1 = 0. Then f(0) is 0 - 3 = -3. Yep, that's exactly what we need. So, the answer is indeed b = -1.

I think that's all. The steps are straightforward, but it's easy to make a sign error or miscalculation. Let me just go through each step again to confirm.

1. Start with f(q(1)) = -3.
2. Compute q(1) = b*1 + 1 = b + 1.
3. Substitute into f: f(b + 1) = (b + 1) - 3 = b - 2.
4. Set equal to -3: b - 2 = -3.
5. Solve: b = -1.

All steps check out. The answer is correct. I don't see any errors here. The key was to substitute step by step and ensure each operation is correctly performed. No complex concepts involved here, just substitution and solving a linear equation. But it's important to be meticulous with signs, especially since the answer is negative. 

Another way to approach this could be by plugging the value back into the original functions. As I did before, substituting b = -1 into q(x) gives a function that when evaluated at x = 1 yields 0, and then applying f to 0 gives -3, which matches the given condition. Therefore, the solution is consistent.

I think that's thorough. The answer is -1.
Therefore, the value of $b$ is $\boxed{-1}$.
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