𝑓(𝑥) = 2𝑥2 + 5𝑥 − 9 ; 𝑓𝑖𝑛𝑑 𝑓(5𝑥 − 2)

To find \( f(5x - 2) \) for the function \( f(x) = 2x^2 + 5x - 9 \), we'll substitute \( 5x - 2 \) for \( x \) in the function.

The original function is:
\[
f(x) = 2x^2 + 5x - 9
\]
Substituting \( 5x - 2 \) for \( x \):
\[
f(5x - 2) = 2(5x - 2)^2 + 5(5x - 2) - 9
\]

Now, let's expand and simplify the expression step by step.

1. **Square \( (5x - 2) \):**
   \[
   (5x - 2)^2 = (5x - 2)(5x - 2) = 25x^2 - 20x + 4
   \]

2. **Multiply the squared term by 2:**
   \[
   2(25x^2 - 20x + 4) = 50x^2 - 40x + 8
   \]

3. **Expand \( 5(5x - 2) \):**
   \[
   5(5x - 2) = 25x - 10
   \]

4. **Combine all the terms:**
   \[
   f(5x - 2) = 50x^2 - 40x + 8 + 25x - 10 - 9
   \]

5. **Simplify the expression:**
   \[
   f(5x - 2) = 50x^2 - 15x - 11
   \]

So, the final expression for \( f(5x - 2) \) is:
\[
\boxed{50x^2 - 15x - 11}
\]

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

Here are some related questions for further exploration:

1. How would you find the derivative of \( f(5x - 2) \) with respect to \( x \)?
2. What is the value of \( f(5x - 2) \) when \( x = 1 \)?
3. How would you graph the function \( f(5x - 2) \)?
4. How does the function \( f(x) \) change when transformed into \( f(5x - 2) \)?
5. How would you find the inverse of the function \( f(x) \)?
6. What are the roots of the equation \( f(5x - 2) = 0 \)?
7. How does scaling and shifting affect the original function \( f(x) \)?
8. What happens to the function \( f(x) \) if you substitute \( x = -2x \) instead of \( 5x - 2 \)?

**Tip:** When substituting expressions into a function, it's important to carefully expand and simplify the expression step by step to avoid errors.

Learn how to find f(5x - 2) for the function f(x) = 2x^2 + 5x - 9 using function composition and quadratic equation principles. This problem covers key algebraic concepts, including quadratic equations and function transformations. Suitable for students in Grades 10-12.

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