https://file.aiquickdraw.com/mathbot/file/17271834483114y5lfel2.jpg

The image shows two problems involving functions and their transformations.

1. The first function is \( f(x) = \frac{2}{5}x + \frac{4}{5} \), and it defines another function, \( g(1)(x) = \frac{-2}{5}x - \frac{4}{5} \). This appears to be the transformation of \( f(x) \) with a negative slope and constant.

2. The second function is \( f_2(x) = \frac{-3}{2}x + \frac{1}{2} \), and the task is to find \( g(2)(x) \), which might be another transformation similar to the first.

Let's find \( g(2)(x) \). If we assume the transformation rule follows the same pattern as in \( g(1)(x) \), it looks like the coefficients of \( f(x) \) are being negated for \( g(x) \). Applying this assumption:

\[
f_2(x) = \frac{-3}{2}x + \frac{1}{2}
\]

Thus, we might define:

\[
g(2)(x) = \frac{3}{2}x - \frac{1}{2}
\]

This mirrors the pattern from \( g(1)(x) \), where the signs of both terms in \( f(x) \) were reversed.

Would you like a deeper explanation of the pattern? Here are some related questions:

1. How does the transformation from \( f(x) \) to \( g(x) \) work in general?
2. What is the geometric interpretation of these transformations?
3. How would \( g(3)(x) \) look for a new function \( f_3(x) \)?
4. Can we express a general formula for any \( g(n)(x) \)?
5. What would happen if \( f(x) \) had higher-degree terms (e.g., \( x^2 \))?

**Tip:** For linear functions, transformations often involve adjusting the slope and intercept. Understanding this concept will help you handle more complex transformations easily.

Given the functions f(x) = 2/5x + 4/5 and f_2(x) = -3/2x + 1/2, find g(1)(x) and g(2)(x) respectively, based on the transformation rule.

Solve a linear function transformation problem where f(x) = 2/5x + 4/5 and f_2(x) = -3/2x + 1/2, using a transformation rule that changes the sign of the slope and constant. Suitable for Grades 9-10.

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