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The problem given is:

For the function \( f(x) = 0.4^x + 1 \), we need to:

1. **Find the domain of the function \( f(x) \)**.
2. **Identify intervals of increase or decrease of \( f(x) \)**.
3. **Determine the range of the function \( f(x) \)**.

### Solution:

#### 1. Domain of the function \( f(x) \)
Since \( f(x) = 0.4^x + 1 \) is defined for all real values of \( x \), there are no restrictions on \( x \) (e.g., no division by zero or square roots of negative numbers).

**Domain:** \( x \in (-\infty, \infty) \).

#### 2. Intervals of Increase or Decrease
To determine the intervals where the function increases or decreases, we examine the base of the exponential function:

- Here, the base is \( 0.4 \), which is less than 1. This means \( f(x) \) is a **decreasing function** since exponential functions with bases between 0 and 1 decay as \( x \) increases.

**Interval of Decrease:** \( x \in (-\infty, \infty) \).

#### 3. Range of the function \( f(x) \)
The exponential function \( 0.4^x \) approaches 0 as \( x \to \infty \) and grows larger (but remains positive) as \( x \to -\infty \).

- The function \( f(x) = 0.4^x + 1 \) will therefore approach 1 from above as \( x \to \infty \), and it will go to infinity as \( x \to -\infty \).

**Range:** \( f(x) \in (1, \infty) \).

Would you like a detailed explanation or have any questions?

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Here are 5 questions that expand on this problem:

1. How would the domain and range change if the function were \( f(x) = -0.4^x + 1 \)?
2. What would the intervals of increase or decrease be if the base of the exponential were greater than 1?
3. How does changing the base of the exponential function affect the shape of its graph?
4. How can we find the asymptotes of the function \( f(x) = 0.4^x + 1 \)?
5. How would transformations like shifts affect the domain and range of an exponential function?

**Tip:** For any exponential function of the form \( f(x) = a \cdot b^x + c \), if \( 0 < b < 1 \), the function is decreasing.

Given the function f(x) = 0.4^x + 1, plot its graph and find:
1. The domain of the function
2. The intervals of increase or decrease
3. The range of the function

Explore the exponential function f(x) = 0.4^x + 1 with a detailed solution, covering domain, range, and intervals of increase or decrease. Suitable for high school students studying exponential functions.

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