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Let's go through each part of the problem step by step.

### (i) Finding the equation for \( g(x) \):
The function \( f(x) = 7x^3 - 8x^2 + 12 \) is given, and you need to write \( g(x) \) such that \( g(x) = -2f(x) - 8 \).

Start by computing \( -2f(x) \):
\[
-2f(x) = -2(7x^3 - 8x^2 + 12) = -14x^3 + 16x^2 - 24
\]

Now, subtract 8:
\[
g(x) = -14x^3 + 16x^2 - 24 - 8 = -14x^3 + 16x^2 - 32
\]

Thus, the equation for \( g(x) \) is:
\[
g(x) = -14x^3 + 16x^2 - 32
\]

### (ii) Writing \( h(x) \) in terms of \( f(x

The function f is given by f(x) = 7x^3 - 8x^2 + 12. Write the equation for polynomial g(x) in standard form such that g(x) = -2f(x) - 8. In the xy-plane, the graph of the function h is the result of translating the graph of the function f by dilating it vertically by a factor of 1/5, then dilating it horizontally by a factor of 4, then vertically translating it down 6 units, and then horizontally translating it right 2 units. Write the function that defines h in terms of f. If the domain and range of function f are D: [2, 8] and R: [10, 20], what are the domain and range of function h?

Solve the problem involving polynomial function g(x) derived from f(x) = 7x^3 - 8x^2 + 12. Learn how to find g(x) and define h(x) based on vertical and horizontal transformations of f(x). Understand the domain and range for both functions.

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