https://file.aiquickdraw.com/mathbot/file/1728783166019f4mf38qz.jpg

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.

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

This problem involves solving the composite function equation g(f(x)) = 0, where f(x) = 2x - 3 and g(x) = x^2. Learn the step-by-step solution to find the largest value of x, involving key algebraic concepts like composite functions and quadratic equations. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation