We are given two functions $f(x) = 2x + 1$ and $(f \circ g)(x) = 3x + 2$. We need to find $g(x)$.

Step-by-Step Solution:

1. Understanding the Composition: Since $(f \circ g)(x) = f(g(x)) = 3x + 2$, we will substitute $g(x)$ into $f(x) = 2x + 1$ and set it equal to $3x + 2$.

2. Setting Up the Equation: Let $g(x) = ax + b$, where $a$ and $b$ are constants to be determined.

3. Substitute $g(x)$ into $f(g(x))$: Using $f(x) = 2x + 1$: $f(g(x)) = f(ax + b) = 2(ax + b) + 1 = 2ax + 2b + 1$

4. Equating with $3x + 2$: We know that $f(g(x)) = 3x + 2$. Thus: $2ax + 2b + 1 = 3x + 2$

5. Solve for $a$ and $b$:

   • By comparing coefficients of $x$, we get $2a = 3$, so $a = \frac{3}{2}$.
   • By comparing the constant terms, we get $2b + 1 = 2$, so $b = \frac{1}{2}$.

6. Conclusion: Thus, $g(x) = \frac{3}{2}x + \frac{1}{2}$.

Answer:

The correct answer is (c) $\frac{3}{2}x + \frac{1}{2}$.

Would you like further details on any of these steps or have any questions?

Related Questions to Explore:

1. How do you find $g(x)$ if $f(g(x))$ and $f(x)$ are given but with different forms?
2. What happens if $f(x)$ is a quadratic function instead of linear?
3. How can you verify your answer using substitution in composite functions?
4. What is the role of inverse functions in such problems?
5. How do you interpret and simplify composite functions with multiple operations?

Tip: When working with composite functions, try substituting variables or expressions directly to keep track of relationships between the functions.