The function \( F(x) = 2^{x+1} + 1 \) can be analyzed as follows:

### Step 1: Simplify the function
The given function is:
\[
F(x) = 2^{x+1} + 1
\]
This can be rewritten by breaking down the exponent:
\[
F(x) = 2^x \cdot 2^1 + 1 = 2 \cdot 2^x + 1
\]
Thus, the function simplifies to:
\[
F(x) = 2 \cdot 2^x + 1
\]

### Step 2: Understand the behavior of the function
- **Exponential Growth:** The term \(2^x\) grows exponentially as \(x\) increases.
- **Shifted Function:** The function is multiplied by 2 and then increased by 1, which shifts the graph upwards.

### Step 3: Analyze the function
- **Domain:** The function is defined for all real numbers, \(x \in \mathbb{R}\).
- **Range:** Since \(2^x\) is always positive, the minimum value of \(F(x)\) occurs as \(x \to -\infty\). However, \(F(x)\) will never reach zero, so the range is \( (1, \infty) \).
- **Intercept:** The y-intercept occurs when \(x = 0\):
  \[
  F(0) = 2^{0+1} + 1 = 2 \cdot 2^0 + 1 = 2 \cdot 1 + 1 = 3
  \]
  So, the function passes through the point \((0, 3)\).

Would you like more details or have any other questions?

Here are some related questions you might consider:
1. What is the derivative of \( F(x) = 2^{x+1} + 1 \)?
2. How does the graph of \( F(x) = 2^{x+1} + 1 \) compare to \( G(x) = 2^x \)?
3. What is the asymptotic behavior of \( F(x) = 2^{x+1} + 1 \)?
4. What is the inverse function of \( F(x) = 2^{x+1} + 1 \)?
5. How does changing the base in the exponent (e.g., \( 3^{x+1} + 1 \)) affect the function?

**Tip:** Understanding how transformations like shifting and stretching affect the graph of a function can make it easier to sketch and interpret.

Explore the function F(x) = 2^{x+1} + 1 with detailed analysis and step-by-step solution. Understand its exponential growth, transformation properties, domain, range, and intercepts. Suitable for Grades 10-12 mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation