Is $$ y=3^{-x} $$ an increasing function or a decreasing function? Dncreasing

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The function $$ y = 3^{-x} $$ is a **decreasing** function. Here's why: ### Understanding the Function 1. **Rewriting the Function:** $$ y = 3^{-x} = \left(\frac{1}{3}\right)^x $$ This shows that the function is an exponential function with a base between 0 and 1. 2. **Behavior of Exponential Functions:** - **Base > 1:** The function increases as $$ x $$ increases. - **0 < Base < 1:** The function decreases as $$ x $$ increases. Since $$ \frac{1}{3} $$ is between 0 and 1, $$ y = \left(\frac{1}{3}\right)^x $$ decreases as $$ x $$ increases. ### Calculating the Derivative To further confirm, let's find the derivative of $$ y $$ with respect to $$ x $$: $$ y = 3^{-x} \ y = e^{-x \ln 3} \ \frac{dy}{dx} = -\ln 3 \cdot e^{-x \ln 3} = -\ln 3 \cdot 3^{-x} $$ Since $$ \ln 3 > 0 $$ and $$ 3^{-x} > 0 $$ for all real $$ x $$, the derivative $$ \frac{dy}{dx} $$ is negative. A negative derivative indicates that the function is decreasing. ### Graphical Interpretation If you plot $$ y = 3^{-x} $$, you'll observe that as $$ x $$ moves from left to right (increasing), the value of $$ y $$ decreases, reinforcing that the function is decreasing. ### Conclusion $$ y = 3^{-x} $$ **is a decreasing function** because as the input $$ x $$ increases, the output $$ y $$ decreases. $$ y = 3^{-x} $$ is a decreasing function.

Is \( y=3^{-x} \) an increasing function or a decreasing function? Dncreasing

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