The functions f and g are defined by f(x)= 3x+2, xer, g(x)=6/2x+3, xer, x≠-1.5, find the value of x for which fg(x)=3

We are given two functions \( f(x) = 3x + 2 \) and \( g(x) = \frac{6}{2x + 3} \), and we need to find the value of \( x \) for which \( f(g(x)) = 3 \).

### Steps:

1. **Substitute \( g(x) \) into \( f(x) \):**

   We want to compute \( f(g(x)) \), which means substituting \( g(x) \) into the function \( f(x) \). Since \( f(x) = 3x + 2 \), replacing \( x \) with \( g(x) \):

   \[
   f(g(x)) = 3 \cdot g(x) + 2
   \]

2. **Express \( f(g(x)) \):**

   Since \( g(x) = \frac{6}{2x + 3} \), substituting this into \( f(g(x)) \):

   \[
   f(g(x)) = 3 \left(\frac{6}{2x + 3}\right) + 2
   \]

   Simplifying this expression:

   \[
   f(g(x)) = \frac{18}{2x + 3} + 2
   \]

3. **Set the equation \( f(g(x)) = 3 \):**

   Now, set the expression for \( f(g(x)) \) equal to 3 and solve for \( x \):

   \[
   \frac{18}{2x + 3} + 2 = 3
   \]

4. **Solve the equation:**

   Subtract 2 from both sides:

   \[
   \frac{18}{2x + 3} = 1
   \]

   Now, solve for \( x \):

   \[
   18 = 2x + 3
   \]
   \[
   2x = 15
   \]
   \[
   x = \frac{15}{2}
   \]

Thus, the value of \( x \) for which \( f(g(x)) = 3 \) is \( x = \frac{15}{2} \) or \( x = 7.5 \).

Would you like a detailed explanation of any step or have further questions? Here are some related problems:

1. Find \( x \) if \( f(g(x)) = 4 \).
2. Solve for \( x \) if \( g(f(x)) = 5 \).
3. Determine the inverse of \( f(x) \) and \( g(x) \).
4. Find the domain of the composite function \( f(g(x)) \).
5. Explore the behavior of \( f(g(x)) \) as \( x \to -1.5 \).

**Tip:** When solving composite functions, always substitute carefully and follow the order of operations precisely to avoid errors.

The functions f and g are defined by f(x) = 3x + 2, x ∈ ℝ, g(x) = 6 / (2x + 3), x ∈ ℝ, x ≠ -1.5. Find the value of x for which f(g(x)) = 3.

Learn how to solve for x in the composite function equation f(g(x)) = 3, where f(x) = 3x + 2 and g(x) = 6 / (2x + 3). This problem covers key algebraic concepts such as composite functions and rational expressions. Suitable for high school students in Grades 9-12.

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