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

Given the functions \( f(x) = ax + b \) and \( g(x) = 1 - x + x^2 \), we need to find \( a \) and \( b \) such that \( (g \circ f)(x) = 9x^2 + 9x + 3 \).

### Step 1: Compute \( g(f(x)) \)
The composition \( (g \circ f)(x) \) means we need to substitute \( f(x) = ax + b \) into \( g(x) \). So, compute \( g(f(x)) \):

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

### Step 2: Expand the expression
Now, expand \( (ax + b)^2 \):

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

Thus, the full expression for \( g(f(x)) \) becomes:

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

Simplify the equation:

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

Group the terms by powers of \( x \):

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

### Step 3: Compare coefficients
We are given that \( g(f(x)) = 9x^2 + 9x + 3 \). Now, compare this expression with the expanded form of \( g(f(x)) \):

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

From this, we can match the coefficients of like terms:
1. Coefficient of \( x^2 \): \( a^2 = 9 \)
2. Coefficient of \( x \): \( 2ab - a = 9 \)
3. Constant term: \( 1 - b + b^2 = 3 \)

### Step 4: Solve the system of equations

1. From \( a^2 = 9 \), we get \( a = 3 \) or \( a = -3 \).
   
2. Using \( a = 3 \) or \( a = -3 \) in the second equation \( 2ab - a = 9 \):

   - If \( a = 3 \), the equation becomes \( 6b - 3 = 9 \), which simplifies to \( 6b = 12 \), so \( b = 2 \).
   - If \( a = -3 \), the equation becomes \( -6b + 3 = 9 \), which simplifies to \( -6b = 6 \), so \( b = -1 \).

3. Substitute the values of \( a \) and \( b \) into the third equation \( 1 - b + b^2 = 3 \):

   - If \( a = 3 \) and \( b = 2 \): 
     \[
     1 - 2 + 2^2 = 1 - 2 + 4 = 3 \quad \text{(True)}
     \]
   - If \( a = -3 \) and \( b = -1 \):
     \[
     1 - (-1) + (-1)^2 = 1 + 1 + 1 = 3 \quad \text{(True)}
     \]

### Step 5: Conclusion
Both pairs \( (a = 3, b = 2) \) and \( (a = -3, b = -1) \) satisfy the equation. Therefore, the possible values for \( a \) and \( b \) are:

\[
(a, b) = (3, 2) \quad \text{or} \quad (a, b) = (-3, -1)
\]

Let me know if you'd like more details or have any questions!

---

Here are 5 related questions you might find interesting:
1. How do we solve more complex function compositions?
2. What are other methods to approach solving function compositions?
3. How do function compositions behave with non-polynomial functions?
4. Can you explain how to decompose a composite function into its components?
5. How does the behavior of a quadratic function change with its coefficients?

**Tip:** When dealing with function compositions, always focus on substituting one function into the other and carefully simplify step-by-step.

Learn how to determine the values of a and b in the function composition problem where f(x) = ax + b, g(x) = 1 - x + x^2, and (g ∘ f)(x) = 9x^2 + 9x + 3. This solution explores key algebraic concepts such as polynomial expansion and coefficient matching. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation