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A logarithmic function like \( f(x) = \log_{1/2}(1-x) \) has specific requirements for its domain and range. The argument of the logarithm, in this case \( 1-x \), must be positive. Hence, for \( f(x) \) to be defined, \( 1-x > 0 \), simplifying to \( x < 1 \). Thus, the domain for this function is all real numbers where \( x < 1 \).
The range of logarithmic functions is generally all real numbers. This is because as \( x \) approaches the lower part of its domain (in this case near 1, just under it), the function value can decrease toward \(-\infty\). As \( x \) decreases further to very negative values, the function value can also increase to very high numbers. Therefore, we can say the range of \( f(x) \) is all real numbers from \(-\infty\) to \( \infty \).
Understanding these constraints is crucial as they affect how we approach graphing the function effectively.

Logarithmic functions often have vertical asymptotes, helping us understand where the graph shoots off towards infinity. In this function, \( f(x) = \log_{1/2}(1-x) \), there is a vertical asymptote where the function becomes undefined. This means looking at the point where \( 1-x = 0 \), which gives \( x = 1 \).
As \( x \) approaches 1 from the left (denoted as \( x \to 1^- \)), the value of \( 1-x \) approaches zero. Since the base of the logarithm \( 1/2 \) is less than 1, \( \log_{1/2}(y) \) diverges toward \(-\infty\) as \( y \to 0^+ \). Hence, the line \( x = 1 \) is a vertical asymptote and the function decreases toward \(-\infty\) as \( x \) approaches this value. Vertical asymptotes are crucial markers when graphing as they determine where the curve extends off the graph without ever touching a certain line.

Graphing \( f(x) = \log_{1/2}(1-x) \) involves using strategic points and understanding the behavior of logarithmic functions. First, determining key x-intercepts like where \( \log_{1/2}(1-x) = 0 \) gives us \( 1-x = 1 \), leading to \( x = 0 \). This x-intercept implies that the curve crosses the x-axis at \( x = 0 \).
To plot the graph accurately, evaluate \( f(x) \) at various points within the domain such as \( f(0) = 0 \), \( f(-1) = \log_{1/2}(2) = -1 \), and \( f(-3) = \log_{1/2}(4) = -2 \). Plotting these points reveals the downward slope as \( x \) goes from left to right, approaching the asymptote. Remember that with \( x \to 1^- \), \( f(x) \rightarrow -\infty \), pulling the graph steeply downward near the asymptote.
Connect these points with a smooth curve keeping the asymptote in mind, reflecting a decrease as x increases toward the vertical asymptote. This provides a holistic graphical representation of the entire function, essential for visualizing how logarithmic functions behave in real contexts.