To solve this problem, let's go through the given information and determine the value of $b$.

We are given:

1. $f(x) = ax + b$
2. $g(x) = 2f(x) - 3$

We know the following values:

• $g(1) = 3$
• $g(3) = 5$

Step 1: Express $g(x)$ in terms of $a$, $b$, and $x$

Since $g(x) = 2f(x) - 3$, we substitute $f(x) = ax + b$ into this equation: $g(x) = 2(ax + b) - 3$ Simplify this expression: $g(x) = 2ax + 2b - 3$

Step 2: Set up equations using $g(1) = 3$ and $g(3) = 5$

1. For $g(1) = 3$: Substitute $x = 1$ into $g(x)$: $g(1) = 2a(1) + 2b - 3 = 3$ Simplify: $2a + 2b - 3 = 3$ $2a + 2b = 6$ $a + b = 3 \quad \text{(Equation 1)}$

2. For $g(3) = 5$: Substitute $x = 3$ into $g(x)$: $g(3) = 2a(3) + 2b - 3 = 5$ Simplify: $6a + 2b - 3 = 5$ $6a + 2b = 8$ Divide by 2: $3a + b = 4 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

We now have the following system of equations:

1. $a + b = 3$
2. $3a + b = 4$

Subtract Equation 1 from Equation 2: $(3a + b) - (a + b) = 4 - 3$ $2a = 1$ $a = \frac{1}{2}$

Now substitute $a = \frac{1}{2}$ back into Equation 1: $\frac{1}{2} + b = 3$ $b = 3 - \frac{1}{2}$ $b = \frac{5}{2}$

Answer

The value of $b$ is: $\boxed{\frac{5}{2}}$

This corresponds to option C.

Would you like a detailed explanation of any step or have further questions?

Here are five related questions to explore this topic further:

1. How would the answer change if $g(1)$ and $g(3)$ had different values?
2. What does the parameter $a$ represent in the function $f(x) = ax + b$?
3. How does the transformation $g(x) = 2f(x) - 3$ affect the graph of $f(x)$?
4. If $f(x)$ were a quadratic function, how would we approach finding $g(x)$?
5. What are the implications of $b$ in determining the intercepts of $f(x)$?

Tip: When solving a system of equations, try isolating one variable in terms of the other to simplify substitution, especially with linear equations.