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Given the problem statement:

We have two functions:
- \( f(x) = 2x - 3 \)
- \( g(x) = x^2 \)

We are tasked to find the largest value of \( x \) such that \( g(f(x)) = 0 \).

### Step-by-Step Solution:

1. **Substitute** \( f(x) \) into \( g(x) \):
   \[
   g(f(x)) = (f(x))^2
   \]
   Since \( f(x) = 2x - 3 \), we substitute:
   \[
   g(f(x)) = (2x - 3)^2
   \]

2. **Set the equation to zero**:
   \[
   (2x - 3)^2 = 0
   \]

3. **Solve for \( x \)**:
   To solve the above equation, we take the square root on both sides:
   \[
   2x - 3 = 0
   \]
   Solving for \( x \):
   \[
   2x = 3
   \]
   \[
   x = \frac{3}{2} = 1.5
   \]

So, the largest value of \( x \) that satisfies the condition \( g(f(x)) = 0 \) is \( x = 1.5 \).

Would you like more details or have any questions about this solution?

### Related Questions:
1. What happens to the value of \( x \) if the condition is \( g(f(x)) > 0 \)?
2. How would you find the smallest value of \( x \) that satisfies \( g(f(x)) = 0 \)?
3. What are the implications if \( g(x) \) was a different function, like a cubic function?
4. How do we determine the range of the function \( g(f(x)) \)?
5. What is the graphical representation of the functions \( f(x) \) and \( g(x) \)?

### Tip:
When solving composite function problems, always start by clearly defining the functions and substituting systematically to avoid errors.

Find the largest value of x if g(f(x)) = 0 where f(x) = 2x - 3 and g(x) = x^2.

Solve the composite function problem where f(x) = 2x - 3 and g(x) = x^2. Learn how to find the largest value of x such that g(f(x)) = 0 using step-by-step substitution and solving techniques. Suitable for students in Grades 8-10.

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