The problem asks us to find the largest value of $x$ for which $g(f(x)) = 0$, where the functions $f(x)$ and $g(x)$ are defined as:

$f(x) = 2x - 3$ $g(x) = x^2$

Step 1: Substitute $f(x)$ into $g(x)$

We need to solve $g(f(x)) = 0$, so first substitute $f(x)$ into $g(x)$:

$g(f(x)) = (f(x))^2 = 0$

Since $f(x) = 2x - 3$, this becomes:

$(2x - 3)^2 = 0$

Step 2: Solve for $x$

Next, solve the equation $(2x - 3)^2 = 0$. To do this, take the square root of both sides:

$2x - 3 = 0$

Solving for $x$:

$2x = 3$ $x = \frac{3}{2}$

Step 3: Conclusion

The largest value of $x$ that satisfies the equation $g(f(x)) = 0$ is:

$x = 1.5$

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

Here are 5 related questions to explore further:

1. What is the general process for solving composite functions like $g(f(x))$?
2. How do we determine if a function has no solutions, multiple solutions, or a single solution?
3. What would happen if $g(x)$ were a different function (e.g., $g(x) = x^3$)?
4. How would the solution change if $f(x)$ had more complex terms?
5. Can $g(f(x)) = 0$ have multiple solutions for different functions?

Tip: When solving composite functions, it's important to carefully substitute one function into the other before attempting to solve the equation.