To solve for $a$ and $b$ in the composite function $(g \circ f)(x) = 9x^2 - 9x + 3$, we first need to understand the definitions and relationships of the functions involved in the composition.

Given:

• $f(x) = 1 - x + x^2$
• $(g \circ f)(x) = 9x^2 - 9x + 3$

We need to determine $g(x)$. Let's assume:

• $f(x) = 1 - x + x^2$
• Let $y = f(x)$
• Then $y = 1 - x + x^2$

We can write $g(y)$ as a function of $y$, where $g(f(x)) = g(y)$.

Given $(g \circ f)(x) = 9x^2 - 9x + 3$, we know that $g(1 - x + x^2) = 9x^2 - 9x + 3$.

Let's assume $g(y) = ay + b$. This is a linear function in terms of $y$.

We know that: $g(1 - x + x^2) = 9x^2 - 9x + 3$

By substituting $y = 1 - x + x^2$ into $g(y)$: $g(1 - x + x^2) = a(1 - x + x^2) + b$ $= a(1 - x + x^2) + b$ $= a(1) + a(-x) + a(x^2) + b$ $= a - ax + ax^2 + b$

We need this to be equal to $9x^2 - 9x + 3$.

So, we compare coefficients: $ax^2 = 9x^2$ $-ax = -9x$ $a + b = 3$

From $ax^2 = 9x^2$: $a = 9$

From $-ax = -9x$: $-9x = -9x$

From $a + b = 3$: $9 + b = 3$ $b = 3 - 9$ $b = -6$

Thus, the values of $a$ and $b$ are: $a = 9$ $b = -6$

So, the function $g(y)$ is: $g(y) = 9y - 6$

Do you have any questions or need further details?

Here are 5 more questions you might be interested in:

1. How do you find the derivative of a composite function?
2. What are the properties of polynomial functions?
3. Can you explain the concept of function composition with more examples?
4. How do you determine the inverse of a function?
5. What are some applications of composite functions in real life?

Tip: When working with composite functions, always ensure to clearly define the inner and outer functions and how they transform inputs step-by-step.