To find the range of the function $y = \left(\frac{1}{5}\right)^x$, we can follow a systematic process.

Step 1: Analyze the Behavior of the Function

The function $y = \left(\frac{1}{5}\right)^x$ is an exponential function where the base is a positive fraction $\frac{1}{5}$. This means the function will behave differently compared to those with bases greater than 1 (like $2^x$ or $3^x$).

Step 2: Identify Key Characteristics of the Function

• As $x \to +\infty$ (i.e., as $x$ increases to large positive values), the value of $\left(\frac{1}{5}\right)^x$ becomes smaller and approaches zero.
• As $x \to -\infty$ (i.e., as $x$ decreases to large negative values), the value of $\left(\frac{1}{5}\right)^x$ increases because the base $\frac{1}{5}$ raised to negative powers becomes larger than 1. Specifically, $\left(\frac{1}{5}\right)^x = 5^{-x}$, and as $x$ gets more negative, $5^{-x} \to \infty$.

Step 3: Examine the Range

• The function approaches 0 as $x \to +\infty$, but never actually reaches 0. It just gets arbitrarily close to 0 from above.
• As $x \to -\infty$, the function grows without bound, approaching infinity.

Therefore, the range of the function $y = \left(\frac{1}{5}\right)^x$ is all positive real numbers greater than 0. In interval notation, this is:

$(0, \infty)$

Conclusion:

The range of the function $y = \left(\frac{1}{5}\right)^x$ is $\boxed{(0, \infty)}$.

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Would you like more details on any part of the solution? Or would you like me to explain any concept further?

Here are some related questions to explore:

1. How does the range change if the base of the exponential function is greater than 1, e.g., $y = 2^x$?
2. What happens to the range if the base is negative, such as $y = (-2)^x$?
3. How does the range of an exponential function differ from a logarithmic function?
4. What is the range of the function $y = 5^x$ and how does it compare to $y = \left(\frac{1}{5}\right)^x$?
5. How can we graph the function $y = \left(\frac{1}{5}\right)^x$ and visualize its range?

Tip: When dealing with exponential functions where the base is between 0 and 1 (like $\frac{1}{5}$), the function will always decrease as $x$ increases, and will approach 0 from above as $x \to +\infty$.