<Question>
Let $f(x) = x - 3$ and $q(x) = bx +1$.  If $f(q(1)) = -3$, what is $b$?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the value of b such that when I compose the functions f and q in the way specified, the result is -3. Hmm. Let me start by understanding the problem step by step.
First, the functions are given as f(x) = x - 3 and q(x) = bx + 1. The condition provided is f(q(1)) = -3. So, I need to compute the composition of f after q at x = 1 and set that equal to -3, then solve for b.

Let me break it down. The function q(x) is a linear function with slope b and y-intercept 1. So, when x is 1, q(1) would be b*1 + 1, which is b + 1.
Then, f(q(1)) means I plug q(1) into f(x). Since f(x) is x - 3, substituting q(1) into f would give f(q(1)) = (q(1)) - 3. Which is (b + 1) - 3.

Let me write that out:

q(1) = b*1 + 1 = b + 1

f(q(1)) = f(b + 1) = (b + 1) - 3 = b + 1 - 3 = b - 2

And according to the problem, this equals -3.

So, setting up the equation:

b - 2 = -3

Solving for b, I just need to add 2 to both sides:

b = -3 + 2 = -1

Wait, that seems straightforward. Let me check if I made any mistakes here.

First, compute q(1): yes, substituting x = 1 into q(x) gives b*1 + 1, which is b + 1.

Then, compute f of that result: f(q(1)) = f(b + 1) = (b + 1) - 3.
Simplify that: b + 1 - 3 = b - 2. Correct.

Set that equal to -3: b - 2 = -3. Solving gives b = -1. Let me verify by plugging back into the original functions.

<prompt_1.1> Verify the solution by plugging the value of \( b \) back into the original functions and checking if the result satisfies the given condition. </prompt_1.1>

Wait, but let me make sure I didn't skip any steps or make any assumptions. Let's re-examine each part.

<prompt_1.2> Re-examine each part of the solution to ensure no steps were skipped and no assumptions were made. </prompt_1.2>

Is there another way to approach this problem? Maybe by composing the functions first and then substituting x = 1? Let's try that.

<prompt_1.3> Explore an alternative method by composing the functions first and then substituting \( x = 1 \) to see if it leads to the same solution. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
If b = -1, then q(x) = -1*x + 1. So q(1) = -1*1 + 1 = 0. Then f(q(1)) = f(0) = 0 - 3 = -3.
</execution_1.1>

<execution_1.2>
First, the definition of function composition. When we write f(q(1)), that means we first apply q to 1, then apply f to the result. So, q(1) is indeed b*1 + 1, which is b + 1. Then, f of that is (b + 1) - 3. Simplify that expression: b + 1 - 3 = b - 2. Then setting that equal to -3: b - 2 = -3. Adding 2 to both sides: b = -1.
</execution_1.2>

<execution_1.3>
The composition f(q(x)) would be f applied to q(x). Since q(x) = bx + 1, then f(q(x)) = f(bx + 1). But f(x) is x - 3, so substituting bx + 1 into f gives (bx + 1) - 3 = bx + 1 - 3 = bx - 2. Therefore, f(q(x)) = bx - 2. Then, evaluating this at x = 1 gives f(q(1)) = b*1 - 2 = b - 2. Setting that equal to -3: b - 2 = -3, so b = -1. Same result.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3:
Alternatively, if I was confused about function composition, I could think of it as f ∘ q (x) = f(q(x)), which we already computed as bx - 2.

<prompt_2.1> Reiterate the solution and confirm the answer through multiple approaches to ensure thorough understanding. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Then evaluating at 1 gives b - 2. So again, same equation.

I don't see any alternative interpretations of the problem. The functions are defined as f(x) = x - 3 and q(x) = bx + 1. The condition is f(q(1)) = -3. The steps all lead to b = -1, and verifying by plugging back in works. Therefore, I think that's the correct answer.

Maybe I can consider if there's a different way someone might approach this problem and make a mistake. For example, someone might confuse the order of composition, thinking it's q(f(1)) instead of f(q(1)). Let's see what happens if someone does that incorrectly.

If it were q(f(1)), then f(1) = 1 - 3 = -2. Then q(-2) = b*(-2) + 1 = -2b + 1. If that were set to -3, then -2b + 1 = -3 → -2b = -4 → b = 2. But that's not the problem here. The problem specifies f(q(1)), so that's not applicable. But just noting that mistake.

Alternatively, someone might miscalculate q(1). If they thought q(x) = b(x + 1), but no, q(x) is given as bx + 1, so it's b*x + 1. So plugging in x =1 gives b*1 +1. Correct.

Another possible mistake: when simplifying f(q(1)), someone might forget to subtract 3. For example, if they thought f(q(1)) is just q(1), which would be b +1, but no, f(x) is x -3, so it's (b +1) -3. So that step must be done carefully.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, the value of $b$ is $\boxed{-1}$.
</Final_answer>
</think>