1-x+x2, where R is the set of all real numbers. If (g o f)(x) =9x2-9x+3, determine a and b, where (g o f) is the composite function.

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.

Explore how to solve for the constants a and b in the composite function (g o f)(x) = 9x^2 - 9x + 3, where f(x) = 1 - x + x^2. Understand the composition of functions, quadratic equations, and linear functions in this detailed mathematical solution.

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