The function given is \( h(x) = 180(0.9)^x \). Let's identify the type of this function and its characteristics.

### 1. **Type of Function:**

The given function \( h(x) = 180(0.9)^x \) is an **exponential function**. It has the general form:

\[
h(x) = a \cdot b^x
\]

where:
- \( a = 180 \) (the initial value or coefficient),
- \( b = 0.9 \) (the base of the exponential function).

### 2. **Characteristics of the Exponential Function:**

- **Base (b):** The base of the exponential function is 0.9, which is less than 1. This indicates that the function is a **decaying exponential** function because the base is between 0 and 1.

- **Initial Value (a):** The initial value is 180, which means that when \( x = 0 \), the function value is 180. This is because \( h(0) = 180 \cdot (0.9)^0 = 180 \).

### 3. **Behavior of the Function:**

- **Exponential Decay:** Since the base \( b = 0.9 \) is less than 1, the function decreases as \( x \) increases. This is known as exponential decay. Each increase in \( x \) reduces \( h(x) \) by 10% of its current value (since \( 0.9 = 1 - 0.1 \), it represents a 10% reduction).

- **Long-Term Behavior:** As \( x \) increases towards infinity, \( h(x) \) approaches 0. This is because \( (0.9)^x \) gets smaller and smaller as \( x \) grows.

### 4. **Domain and Range:**

- **Domain:** The domain of this exponential function is all real numbers, \( \mathbb{R} \). \( x \) can take any real value.

- **Range:** The range of the function is \( (0, \infty) \) since \( h(x) \) is always positive but can get arbitrarily close to zero without ever reaching it.

### 5. **Intercepts:**

- **Y-Intercept:** The y-intercept occurs when \( x = 0 \). For this function:

  \[
  h(0) = 180 \cdot (0.9)^0 = 180 \cdot 1 = 180
  \]

  So, the y-intercept is \( (0, 180) \).

- **X-Intercept:** There is no x-intercept for this function because \( h(x) = 180(0.9)^x \) never equals zero for any real value of \( x \). The function only approaches zero asymptotically.

### Summary:

- **Function Type:** Exponential Decay
- **Initial Value (a):** 180
- **Base (b):** 0.9 (indicating decay)
- **Domain:** All real numbers, \( \mathbb{R} \)
- **Range:** \( (0, \infty) \)
- **Y-Intercept:** \( (0, 180) \)
- **X-Intercept:** None

Would you like more details or have any questions?

### Related Questions:

1. How does changing the base \( b \) to a value greater than 1 affect an exponential function?
2. What are some real-world examples of exponential decay?
3. How can you find the half-life of a substance using an exponential decay function?
4. How does the initial value \( a \) affect the graph of an exponential function?
5. What is the significance of the y-intercept in an exponential function graph?

### Tip:
For exponential decay functions, the base \( b \) is always between 0 and 1. The closer \( b \) is to 0, the faster the decay. The closer \( b \) is to 1, the slower the decay.

Explore the function h(x) = 180 * (0.9)^x, an exponential decay model. Learn about its characteristics, including the base (0.9), initial value (180), domain (all real numbers), range (positive values approaching zero), and intercepts. Suitable for advanced high school students and anyone interested in exponential 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