F(x)=ax+b and g(x)=1-x+x^2 and if gof=9x^2+9x+3 determine a and b

We are given the following functions and their composition:

- \( f(x) = ax + b \)
- \( g(x) = 1 - x + x^2 \)
- \( g(f(x)) = 9x^2 + 9x + 3 \)

The goal is to determine \( a \) and \( b \).

### Step 1: Compute \( g(f(x)) \)
First, substitute \( f(x) = ax + b \) into \( g(x) \). Since \( g(x) = 1 - x + x^2 \), we get:

\[
g(f(x)) = 1 - (ax + b) + (ax + b)^2
\]

Now expand each term:

\[
g(f(x)) = 1 - ax - b + (ax + b)^2
\]

For the square term \( (ax + b)^2 \), use the formula \( (p + q)^2 = p^2 + 2pq + q^2 \):

\[
(ax + b)^2 = a^2x^2 + 2abx + b^2
\]

Substituting this back into the equation for \( g(f(x)) \), we have:

\[
g(f(x)) = 1 - ax - b + a^2x^2 + 2abx + b^2
\]

Simplify the expression:

\[
g(f(x)) = a^2x^2 + (2ab - a)x + (1 - b + b^2)
\]

### Step 2: Set \( g(f(x)) \) equal to \( 9x^2 + 9x + 3 \)
We are told that \( g(f(x)) = 9x^2 + 9x + 3 \). So, equating the expressions for \( g(f(x)) \):

\[
a^2x^2 + (2ab - a)x + (1 - b + b^2) = 9x^2 + 9x + 3
\]

### Step 3: Compare coefficients
Now, compare the coefficients of \( x^2 \), \( x \), and the constant terms on both sides:

1. Coefficient of \( x^2 \):  
   \( a^2 = 9 \)  
   Solving this gives \( a = 3 \) or \( a = -3 \).

2. Coefficient of \( x \):  
   \( 2ab - a = 9 \)

3. Constant term:  
   \( 1 - b + b^2 = 3 \)

### Step 4: Solve for \( b \)
Let’s first solve for \( b \) by examining the constant term equation:

\[
1 - b + b^2 = 3
\]
\[
b^2 - b - 2 = 0
\]

This is a quadratic equation, and we can solve it using the quadratic formula:

\[
b = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}
\]
\[
b = \frac{1 \pm \sqrt{1 + 8}}{2}
\]
\[
b = \frac{1 \pm 3}{2}
\]

So, \( b = 2 \) or \( b = -1 \).

### Step 5: Solve for \( a \)
Now, let’s substitute these values of \( b \) into the equation for the coefficient of \( x \):

\[
2ab - a = 9
\]

#### Case 1: \( a = 3 \)
- If \( b = 2 \):
  \[
  2(3)(2) - 3 = 9 \quad \text{(True)}
  \]
  So, \( a = 3 \) and \( b = 2 \) is a valid solution.

- If \( b = -1 \):
  \[
  2(3)(-1) - 3 = -9 \quad \text{(False)}
  \]
  So, \( b = -1 \) does not work with \( a = 3 \).

#### Case 2: \( a = -3 \)
-

F(x) = ax + b and g(x) = 1 - x + x^2, and if gof = 9x^2 + 9x + 3, determine a and b.

Learn how to solve a function composition problem involving f(x) = ax + b and g(x) = 1 - x + x^2. Determine the values of a and b given that g(f(x)) equals 9x^2 + 9x + 3. This solution includes key algebraic concepts, function composition, and quadratic equation solving techniques. Suitable for students in Grades 10-12.

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