We are given the following functions:

• $f(x) = x - 2$
• $g(x) = ax + b$

Also, we know that:

1. $f \circ g(2) = -3$
2. $g \circ f(1) = 5$

We need to find the values of $a$ and $b$. Let's solve this step by step.

Step 1: Use $f \circ g(2) = -3$

This means we first apply $g$ to 2, and then apply $f$ to the result of $g(2)$:

$f(g(2)) = -3$

Now, substitute $g(x) = ax + b$:

$g(2) = a(2) + b = 2a + b$

Next, apply $f$ to $g(2)$:

$f(2a + b) = (2a + b) - 2$

We are told that $f(g(2)) = -3$, so:

$(2a + b) - 2 = -3$

Solving this equation:

$2a + b - 2 = -3$ $2a + b = -1 \quad \text{(Equation 1)}$

Step 2: Use $g \circ f(1) = 5$

This means we first apply $f$ to 1, and then apply $g$ to the result of $f(1)$:

$g(f(1)) = 5$

Now, find $f(1)$:

$f(1) = 1 - 2 = -1$

Next, apply $g$ to $f(1) = -1$:

$g(-1) = a(-1) + b = -a + b$

We are told that $g(f(1)) = 5$, so:

$-a + b = 5 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

We now have the following system of equations:

1. $2a + b = -1$
2. $-a + b = 5$

Let's solve this system.

From Equation 2, solve for $b$:

$b = a + 5$

Substitute this into Equation 1:

$2a + (a + 5) = -1$ $3a + 5 = -1$ $3a = -6$ $a = -2$

Now substitute $a = -2$ into the equation $b = a + 5$:

$b = -2 + 5 = 3$

Final Answer:

• $a = -2$
• $b = 3$

Let me know if you need more details or clarifications!

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Here are some related questions for further exploration:

1. What is the composite function $f \circ g(x)$?
2. How do you solve systems of linear equations algebraically?
3. What is the importance of composition in function theory?
4. Can the values of $a$ and $b$ be interpreted geometrically?
5. How would you approach a similar problem where more compositions are involved?

Tip: When solving a system of equations, always try to express one variable in terms of the other to simplify the process.