f(x) = ax+b 
g(x) = x^2+x+3

find the two possible functions f such that (g o f)(x)=4x^2-14x+15

We are given two functions:

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

We need to find the two possible functions \( f(x) \) such that:

\[
(g \circ f)(x) = 4x^2 - 14x + 15
\]

This means \( g(f(x)) = 4x^2 - 14x + 15 \). To solve this, we'll first substitute \( f(x) = ax + b \) into \( g(x) \), and then match the resulting expression to \( 4x^2 - 14x + 15 \).

### Step 1: Substituting \( f(x) \) into \( g(x) \)
Since \( g(x) = x^2 + x + 3 \), we substitute \( f(x) = ax + b \) into \( g \):

\[
g(f(x)) = g(ax + b) = (ax + b)^2 + (ax + b) + 3
\]

### Step 2: Expanding the expression
Now, expand each term:

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

So,

\[
g(f(x)) = a^2x^2 + 2abx + b^2 + ax + b + 3
\]

Combine like terms:

\[
g(f(x)) = a^2x^2 + (2ab + a)x + (b^2 + b + 3)
\]

### Step 3: Matching with \( 4x^2 - 14x + 15 \)
We now match the expression \( g(f(x)) = a^2x^2 + (2ab + a)x + (b^2 + b + 3) \) to \( 4x^2 - 14x + 15 \).

By comparing coefficients, we get the following system of equations:

1. \( a^2 = 4 \) (coefficient of \( x^2 \))
2. \( 2ab + a = -14 \) (coefficient of \( x \))
3. \( b^2 + b + 3 = 15 \) (constant term)

### Step 4: Solving the system of equations

#### Equation 1: Solve for \( a \)
From \( a^2 = 4 \), we get two possible values for \( a \):

\[
a = 2 \quad \text{or} \quad a = -2
\]

#### Equation 2: Solve for \( b \)
For each value of \( a \), we substitute into \( 2ab + a = -14 \).

- For \( a = 2 \):

\[
2(2)b + 2 = -14 \quad \Rightarrow \quad 4b + 2 = -14 \quad \Rightarrow \quad 4b = -16 \quad \Rightarrow \quad b = -4
\]

- For \( a = -2 \):

\[
2(-2)b + (-2) = -14 \quad \Rightarrow \quad -4b - 2 = -14 \quad \Rightarrow \quad -4b = -12 \quad \Rightarrow \quad b = 3
\]

#### Equation 3: Check constant term
Now check whether these values satisfy the third equation \( b^2 + b + 3 = 15 \).

- For \( a = 2 \) and \( b = -4 \):

\[
(-4)^2 + (-4) + 3 = 16 - 4 + 3 = 15 \quad \text{(True)}
\]

- For \( a = -2 \) and \( b = 3 \):

\[
3^2 + 3 + 3 = 9 + 3 + 3 = 15 \quad \text{(True)}
\]

### Step 5: Final result
Thus, the two possible functions \( f(x) \) are:

1. \( f(x) = 2x - 4 \)
2. \( f(x) = -2x + 3 \)

Would you like more details or have any questions? 

Here are 5 related questions:

1. What is the composition of functions and how is it applied in this problem?
2. How do we expand the square of a binomial?
3. Can you explain why we compare coefficients when solving such equations?
4. How would the solution change if the quadratic part of \( g(x) \) were different?
5. What are the general steps to solve problems involving function composition and matching coefficients?

**Tip**: When solving for coefficients, it's helpful to carefully organize terms by their powers to avoid errors during comparison.

f(x) = ax+b, g(x) = x^2 + x + 3. Find the two possible functions f such that (g o f)(x) = 4x^2 - 14x + 15.

Explore how to find the two possible functions f(x) in the composition problem g(f(x)) = 4x^2 - 14x + 15, involving algebraic concepts like function composition, polynomial expansion, and solving systems of equations. Suitable for high school students (Grades 10-12).

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