https://file.aiquickdraw.com/mathbot/file/1726859054440kg83vc8e.jpg

The image provides the following piecewise functions for \( f(x) \) and \( g(x) \):

\[
f(x) = 
\begin{cases} 
2x - 1 & \text{for } x < 5 \\
x^2 - 1 & \text{for } x > 5 
\end{cases}
\]

\[
g(x) = 
\begin{cases} 
4x + 3 & \text{for } x < 2 \\
3x - 1 & \text{for } x \geq 2
\end{cases}
\]

The question asks for \( f(x) + 2g(x) \), which involves summing the expressions for \( f(x) \) and \( 2g(x) \) for different ranges of \( x \).

### Step-by-step:

1. **For \( x < 2 \):**
   - \( f(x) = 2x - 1 \)
   - \( g(x) = 4x + 3 \)
   - Therefore, \( f(x) + 2g(x) = (2x - 1) + 2(4x + 3) = 2x - 1 + 8x + 6 = 10x + 5 \)

2. **For \( 2 \leq x < 5 \):**
   - \( f(x) = 2x - 1 \)
   - \( g(x) = 3x - 1 \)
   - Therefore, \( f(x) + 2g(x) = (2x - 1) + 2(3x - 1) = 2x - 1 + 6x - 2 = 8x - 3 \)

3. **For \( x > 5 \):**
   - \( f(x) = x^2 - 1 \)
   - \( g(x) = 3x - 1 \)
   - Therefore, \( f(x) + 2g(x) = (x^2 - 1) + 2(3x - 1) = x^2 - 1 + 6x - 2 = x^2 + 6x - 3 \)

### Final piecewise result:
\[
f(x) + 2g(x) = 
\begin{cases} 
10x + 5 & \text{for } x < 2 \\
8x - 3 & \text{for } 2 \leq x < 5 \\
x^2 + 6x - 3 & \text{for } x > 5
\end{cases}
\]

Would you like more details on any step?

Here are 5 questions you could explore next:
1. How would the expression change if the ranges of \( f(x) \) and \( g(x) \) overlapped differently?
2. What happens to the behavior of \( f(x) + 2g(x) \) as \( x \) approaches 2 from the left and right?
3. How would \( f(x) + 2g(x) \) be impacted if the coefficients of \( g(x) \) were modified?
4. How can we graph this piecewise function for better visualization?
5. What if the function were evaluated at specific points like \( x = 1 \), \( x = 3 \), or \( x = 6 \)?

**Tip:** When working with piecewise functions, always check the boundary conditions carefully to ensure continuity or note any discontinuities.

Explore how to calculate f(x) + 2g(x) where f and g are piecewise functions. This guide includes formulas and concepts like the superposition principle, tailored for high school students studying advanced algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation