To find the function \( f(x) \) given that \( f\left(\frac{1}{2}x\right) = x^2 \), we can use substitution to express \( f(x) \) in terms of \( x \).

Let's define a new variable:

\[
u = \frac{1}{2}x \implies x = 2u
\]

Substitute this back into the equation:

\[
f(u) = (2u)^2 = 4u^2
\]

Since \( u = \frac{1}{2}x \), we replace \( u \) with \( x \):

\[
f(x) = 4x^2
\]

Thus, the function is:

\[
\boxed{f(x) = 4x^2}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How would you find \( f(x) \) if given \( f(2x) = x^2 \)?
2. What if \( f\left(\frac{1}{3}x\right) = x^2 \); what is \( f(x) \)?
3. How do transformations affect the graphs of functions?
4. Can you explain how to determine a function given a composite function?
5. How would you find the inverse of a function like \( f(x) = 4x^2 \)?

**Tip:** When dealing with functions, consider substitutions to simplify expressions or find relationships.

Explore how to find the function f(x) given f(½x) = x² through substitution techniques. Learn about quadratic transformations and function composition, suitable for students in Grades 9-11.

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