9. \( f(x)=0.2(5)^{-x} \)

Sure, let's analyze the function you've provided: \[ f(x) = 0.2 \cdot (5)^{-x} \] ### 1. **Understanding the Function** The function \( f(x) = 0.2 \cdot (5)^{-x} \) is an **exponential function**. To better understand its behavior, let's rewrite it in a more familiar exponential form: \[ f(x) = 0.2 \cdot \left(\frac{1}{5}\right)^x \] This is because \( 5^{-x} = \left(\frac{1}{5}\right)^x \). ### 2. **Key Characteristics** #### a. **Base of the Exponential Function** - **Base (\( b \))**: \( \frac{1}{5} \) or \( 0.2 \) - Since \( 0 < b < 1 \), this indicates **exponential decay**. #### b. **Initial Value (y-intercept)** To find the y-intercept, set \( x = 0 \): \[ f(0) = 0.2 \cdot \left(\frac{1}{5}\right)^0 = 0.2 \cdot 1 = 0.2 \] So, the y-intercept is at \( (0, 0.2) \). #### c. **Horizontal Asymptote** As \( x \) approaches \( +\infty \), \( \left(\frac{1}{5}\right)^x \) approaches 0. Therefore, the horizontal asymptote is: \[ y = 0 \] #### d. **Behavior of the Function** - **As \( x \) increases (\( x \to +\infty \))**: \[ f(x) \to 0 \] - **As \( x \) decreases (\( x \to -\infty \))**: \[ f(x) \to \infty \] This means the function decreases towards the horizontal asymptote as \( x \) increases and increases without bound as \( x \) decreases. #### e. **Domain and Range** - **Domain**: All real numbers (\( -\infty, +\infty \)) - **Range**: All positive real numbers (\( 0, +\infty \)) ### 3. **Graph of the Function** Here's a qualitative description of the graph: - **Y-intercept** at \( (0, 0.2) \). - The graph approaches the horizontal asymptote \( y = 0 \) as \( x \) increases. - The graph rises steeply as \( x \) becomes negative. - It is continuously decreasing throughout its domain. **Graph Sketch:** ``` y | | | | * | * | * | * | * |* |-------------------------- x ``` *(Note: The asterisks (*) represent points on the graph approaching the y-axis as \( x \) decreases and approaching the horizontal asymptote \( y = 0 \) as \( x \) increases.)* ### 4. **Additional Analysis** #### a. **Inverse Function** To find the inverse of \( f(x) \), solve for \( x \): \[ y = 0.2 \cdot \left(\frac{1}{5}\right)^x \] First, divide both sides by 0.2: \[ \frac{y}{0.2} = \left(\frac{1}{5}\right)^x \] Take the natural logarithm of both sides: \[ \ln\left(\frac{y}{0.2}\right) = x \cdot \ln\left(\frac{1}{5}\right) \] Solve for \( x \): \[ x = \frac{\ln\left(\frac{y}{0.2}\right)}{\ln\left(\frac{1}{5}\right)} \] So, the inverse function is: \[ f^{-1}(y) = \frac{\ln\left(\frac{y}{0.2}\right)}{\ln\left(\frac{1}{5}\right)} \] #### b. **Applications** Exponential decay functions like this one are commonly used to model: - **Radioactive decay** - **Depreciation of assets** - **Cooling of objects** Understanding the parameters helps in interpreting real-world scenarios where quantities decrease exponentially over time. ### 5. **Summary** - **Function Type**: Exponential decay - **Base**: \( \frac{1}{5} \) (since it's between 0 and 1) - **Y-intercept**: \( 0.2 \) - **Horizontal Asymptote**: \( y = 0 \) - **Domain**: All real numbers - **Range**: Positive real numbers If you have a specific question or need further analysis on this function, feel free to ask!