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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?

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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.

In the functions f and g above, if g(1) = 3 and g(3) = 5, what is the value of b?

Learn how to solve for b in the linear function transformation problem where g(x) = 2f(x) - 3 and specific values are given for g(1) and g(3). This step-by-step solution includes setting up and solving a system of equations, suitable for high school students in Grades 9-11.

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