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.