To graph the function \( f(x) = \left(\frac{1}{2}\right)^x \), we start by evaluating the function at various values of \( x \). This process involves substituting different \( x \) values into the function to find corresponding \( y \) values, which we can then plot on a graph.

First, when \( x = 0 \), we calculate:

\[ f(0) = \left(\frac{1}{2}\right)^0 = 1 \]

This gives us our first point at \( (0, 1) \).

Next, for \( x = 1 \):

\[ f(1) = \left(\frac{1}{2}\right)^1 = \frac{1}{2} \]

We plot the second point at \( (1, \frac{1}{2}) \).

Continuing with \( x = 2 \):

\[ f(2) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

This gives us the point \( (2, \frac{1}{4}) \).

For \( x = 3 \):

\[ f(3) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

We plot the point \( (3, \frac{1}{8}) \), which is getting closer to the x-axis.

Now, we explore negative values of \( x \). Starting with \( x = -1 \):

\[ f(-1) = \left(\frac{1}{2}\right)^{-1} = 2 \]

This gives us the point \( (-1, 2) \).

For \( x = -2 \):

\[ f(-2) = \left(\frac{1}{2}\right)^{-2} = 4 \]

We plot the point \( (-2, 4) \).

Finally, for \( x = -3 \):

\[ f(-3) = \left(\frac{1}{2}\right)^{-3} = 8 \]

This results in the point \( (-3, 8) \).

As we observe the points plotted, we notice that for positive \( x \), the function values decrease towards zero, while for negative \( x \), the function values increase significantly. This behavior indicates that the graph approaches the x-axis but never touches it, suggesting a horizontal asymptote at \( y = 0 \).

It is also important to note that the function \( f(x) = \left(\frac{1}{2}\right)^x \) can be rewritten as \( f(x) = 2^{-x} \). This transformation shows that the graph of \( \left(\frac{1}{2}\right)^x \) is a reflection of the graph of \( 2^x \) across the y-axis.

In terms of the domain and range of the function, the domain of all exponential functions is all real numbers, represented as \( (-\infty, \infty) \). The range, determined by the asymptote, is from \( 0 \) to \( \infty \), expressed as \( (0, \infty) \), indicating that the function never reaches zero but can get infinitely close.

In summary, the graph of \( f(x) = \left(\frac{1}{2}\right)^x \) exhibits exponential decay, characterized by a horizontal asymptote at \( y = 0 \), a domain of all real numbers, and a range of positive values extending to infinity.