The function g is defined by f such that g(x) = 3f(ax)+1 if(x) =x^2-3x and g(2) =1 what is the value for all a>0?

We are given two functions and a specific condition. Let's break this down to find the value of \( a \) when \( a > 0 \).

### Step 1: Function Definition
- We are told that \( g(x) = 3f(ax) + 1 \).
- Additionally, we know \( f(x) = x^2 - 3x \).

Thus, the function \( g(x) \) becomes:
\[
g(x) = 3f(ax) + 1 = 3\left( (ax)^2 - 3(ax) \right) + 1 = 3\left( a^2x^2 - 3ax \right) + 1.
\]
Simplifying this expression:
\[
g(x) = 3a^2x^2 - 9ax + 1.
\]

### Step 2: Use the Condition \( g(2) = 1 \)
We are also given that \( g(2) = 1 \). Substituting \( x = 2 \) into the equation for \( g(x) \):
\[
g(2) = 3a^2(2)^2 - 9a(2) + 1 = 1.
\]
Simplifying this:
\[
g(2) = 3a^2(4) - 18a + 1 = 1,
\]
\[
12a^2 - 18a + 1 = 1.
\]
Subtract 1 from both sides:
\[
12a^2 - 18a = 0.
\]

### Step 3: Solve the Equation for \( a \)
Factor the equation:
\[
6a(2a - 3) = 0.
\]
This gives two possible solutions:
\[
a = 0 \quad \text{or} \quad a = \frac{3}{2}.
\]

### Step 4: Consider \( a > 0 \)
Since we are told that \( a > 0 \), we reject \( a = 0 \) and conclude that:
\[
a = \frac{3}{2}.
\]

Thus, the value of \( a \) is \( \frac{3}{2} \) for all \( a > 0 \).

Would you like further details or have any questions about this solution?

Here are 5 related questions to expand your understanding:
1. How do you factor a quadratic expression like \( 12a^2 - 18a = 0 \)?
2. Can you explain why \( a = 0 \) is not a valid solution in this case?
3. How would the solution change if \( f(x) \) was a different quadratic function?
4. What happens to \( g(x) \) if \( f(x) \) has an added constant term?
5. Can you derive the general form of \( g(x) \) for any function \( f(x) \)?

**Tip:** Always verify your solution by substituting it back into the original equation to check its validity.

Explore how to solve a quadratic function composition problem where g(x) = 3f(ax) + 1 with f(x) = x^2 - 3x, and the condition g(2) = 1. This solution involves key algebraic concepts such as quadratic equations and factoring. Ideal for high school students in Grades 9-12.

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