27) \( y=\left(\frac{1}{2}\right)^{x} \)

Certainly! Let's analyze the exponential function: \[ y = \left(\frac{1}{2}\right)^{x} \] This function is an example of an exponential decay function. Here's a detailed breakdown of its properties and behavior. ## 1. **General Form of an Exponential Function** The general form of an exponential function is: \[ y = a \cdot b^{x} \] Where: - \( a \) is the initial value (the y-intercept). - \( b \) is the base of the exponential function. - If \( 0 < b < 1 \), the function represents exponential decay. - If \( b > 1 \), the function represents exponential growth. In our case: - \( a = 1 \) (since there's no coefficient other than 1). - \( b = \frac{1}{2} \), which is between 0 and 1, indicating exponential decay. ## 2. **Domain and Range** - **Domain:** All real numbers (\( -\infty, \infty \)) - **Range:** \( y > 0 \) (since any positive base raised to a real exponent is positive) ## 3. **Asymptotes** - **Horizontal Asymptote:** \( y = 0 \) As \( x \) approaches \( +\infty \), \( y \) approaches 0. - **No Vertical Asymptotes** ## 4. **Intercepts** - **Y-intercept:** Occurs when \( x = 0 \) \[ y = \left(\frac{1}{2}\right)^{0} = 1 \] So, the y-intercept is at \( (0, 1) \). - **X-intercept:** An exponential function of this form never crosses the x-axis because \( y \) never equals zero. Thus, there is no x-intercept. ## 5. **Behavior and Graph** - **For \( x > 0 \):** The function decreases towards 0 as \( x \) increases. For example: \[ y = \left(\frac{1}{2}\right)^{1} = \frac{1}{2} \] \[ y = \left(\frac{1}{2}\right)^{2} = \frac{1}{4} \] \[ y = \left(\frac{1}{2}\right)^{3} = \frac{1}{8} \] - **For \( x < 0 \):** The function increases without bound as \( x \) becomes more negative. For example: \[ y = \left(\frac{1}{2}\right)^{-1} = 2 \] \[ y = \left(\frac{1}{2}\right)^{-2} = 4 \] \[ y = \left(\frac{1}{2}\right)^{-3} = 8 \] ## 6. **Graph of the Function** Below is a sketch of the graph of \( y = \left(\frac{1}{2}\right)^{x} \): ``` y | | * | * | * | * | * | * | * | * | * | * |-----*-----------*-----------*----- x -3 0 3 ``` - The graph approaches the horizontal asymptote \( y = 0 \) as \( x \) increases. - It passes through the y-intercept at \( (0, 1) \). - For negative values of \( x \), the graph increases rapidly. ## 7. **Applications** Exponential decay functions like \( y = \left(\frac{1}{2}\right)^{x} \) are commonly used to model situations where a quantity decreases rapidly at first and then levels off over time. Examples include: - Radioactive decay - Depreciation of assets - Cooling of an object - Population decline under certain conditions ## 8. **Inverse Function** The inverse of the exponential function \( y = \left(\frac{1}{2}\right)^{x} \) is the logarithmic function: \[ x = \log_{\frac{1}{2}}(y) \] This can also be written using natural logarithms: \[ x = \frac{\ln(y)}{\ln\left(\frac{1}{2}\right)} = -\frac{\ln(y)}{\ln(2)} \] ## 9. **Differential and Integral Calculations** - **Derivative:** The derivative of \( y = \left(\frac{1}{2}\right)^{x} \) with respect to \( x \) is: \[ \frac{dy}{dx} = \left(\frac{1}{2}\right)^{x} \ln\left(\frac{1}{2}\right) \] Since \( \ln\left(\frac{1}{2}\right) = -\ln(2) \), the derivative simplifies to: \[ \frac{dy}{dx} = -\ln(2) \left(\frac{1}{2}\right)^{x} \] This negative derivative confirms that the function is decreasing. - **Integral:** The indefinite integral of \( y = \left(\frac{1}{2}\right)^{x} \) with respect to \( x \) is: \[ \int \left(\frac{1}{2}\right)^{x} dx = \frac{\left(\frac{1}{2}\right)^{x}}{\ln\left(\frac{1}{2}\right)} + C = -\frac{\left(\frac{1}{2}\right)^{x}}{\ln(2)} + C \] Where \( C \) is the constant of integration. ## 10. **Summary of Key Points** - **Type:** Exponential decay function - **Base:** \( \frac{1}{2} \) (between 0 and 1) - **Y-intercept:** \( (0, 1) \) - **Horizontal Asymptote:** \( y = 0 \) - **Behavior:** Decreases towards 0 as \( x \) increases; increases without bound as \( x \) decreases - **Applications:** Modeling decay processes, depreciation, cooling, etc. Understanding the properties of exponential functions is fundamental in various fields such as mathematics, physics, economics, and biology, where growth and decay processes are studied.