https://file.aiquickdraw.com/mathbot/file/1726419578397ess75lu6.jpg

The image contains the following information:

1. The function is given as:
   \[
   f(x) = ax^2 + bx + 3 \quad \text{for} \quad x \geq 2
   \]

2. Two specific function values are provided:
   \[
   f(2) = 3
   \]
   \[
   f(f(3)) = 4.5
   \]

### Task:
- We need to find the constants \(a\) and \(b\), and verify the given values of \(f(x)\).

### Step 1: Use \(f(2) = 3\) to find the first equation
Substitute \(x = 2\) into the function expression:
\[
f(2) = a(2)^2 + b(2) + 3 = 4a + 2b + 3
\]
We are given \(f(2) = 3\), so:
\[
4a + 2b + 3 = 3 \implies 4a + 2b = 0 \quad \text{(Equation 1)}
\]

### Step 2: Use \(f(f(3)) = 4.5\) to find another equation
First, find \(f(3)\):
\[
f(3) = a(3)^2 + b(3) + 3 = 9a + 3b + 3
\]
Now, substitute \(f(3)\) into \(f(x)\):
\[
f(f(3)) = f(9a + 3b + 3)
\]
Given that \(f(f(3)) = 4.5\), we'll need to solve this.

Let's calculate these values.We have:

1. From \( f(2) = 3 \), the equation is:
   \[
   4a + 2b = 0 \quad \text{(Equation 1)}
   \]

2. The expression for \( f(3) \) is:
   \[
   f(3) = 9a + 3b + 3
   \]

To proceed, we can use \( f(f(3)) = 4.5 \) by substituting this into the function. First, let's substitute \( f(3) = 9a + 3b + 3 \) into the function expression and solve using the given condition \( f(f(3)) = 4.5 \).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you need further clarification, feel free to ask. Would you like to proceed with the next steps manually or discuss any details further?

Here are 5 related questions to deepen your understanding:
1. How do we verify if a given function value is correct using a specific equation?
2. What are the general steps to solve quadratic functions for unknown coefficients?
3. Can you derive another example where two function values are given to find constants?
4. What is the significance of solving for \(f(f(x))\) in a composition of functions?
5. How does the constraint \(x \geq 2\) affect the function's domain?

**Tip**: When dealing with nested functions like \(f(f(x))\), simplify the inner function first before substitution to avoid unnecessary complexity.

Given the function f(x) = ax^2 + bx + 3 for x >= 2, with f(2) = 3 and f(f(3)) = 4.5, find the values of a and b.

Learn how to solve for the constants a and b in the quadratic function f(x) = ax^2 + bx + 3, given specific values f(2) = 3 and f(f(3)) = 4.5. This problem covers important algebraic concepts like function composition and solving quadratic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation