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The domain of a function specifies the set of allowable input values, namely the x-values for which the function is well-defined. In the case of the function \( f(x) = \frac{1}{x} - 4 \), the fraction within the function becomes undefined at any point where the denominator is zero. Therefore, for this function, we exclude \( x = 0 \) from the domain, since division by zero is not feasible. As such, the domain of this function is all real numbers except zero, which we denote as \( \mathbb{R} \setminus \{0\} \) or in interval notation, \( (-\infty, 0) \cup (0, \infty) \).
To understand the range, we focus on the possible output values, or y-values. For \( f(x) = \frac{1}{x} - 4 \), as \( x \) approaches either positive or negative infinity, the term \( \frac{1}{x} \) tends towards zero. Hence, \( f(x) \) tends towards \( -4 \) but never actually attains this value as \( \frac{1}{x} \) never becomes exactly zero. Therefore, the range is all real numbers except \( -4 \), which can be expressed as \( \mathbb{R} \setminus \{-4\} \).
Understanding these sets helps us to analyze and predict the behavior of the function effectively.

Asymptotes are lines that a graph approaches but never actually touches or crosses. These can be vertical or horizontal, depending on the properties of the function.
For the function \( f(x) = \frac{1}{x} - 4 \), identifying the vertical asymptote involves determining where the function is undefined. Since \( f(x) \) is undefined when \( x = 0 \), there is a vertical asymptote at \( x = 0 \).
A horizontal asymptote describes the behavior of a curve as \( x \) approaches infinity or negative infinity. By calculating \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \), we find that \( f(x) \) tends towards \( -4 \). This result presents a horizontal asymptote at \( y = -4 \).
Understanding these asymptotes is crucial, as they map out the territory beyond which the graph does not extend, providing insight into the end-behavior of the function.

Intercepts reveal the points where the graph crosses the axes. In the context of \( f(x) = \frac{1}{x} - 4 \), identifying intercepts helps us explore significant behaviors of the function.
To locate the y-intercept, one would typically set \( x = 0 \). However, this function is undefined at \( x = 0 \), and therefore possesses no y-intercept.
Finding the x-intercept requires setting the entire function equal to zero: \( \frac{1}{x} - 4 = 0 \). Solving implies \( \frac{1}{x} = 4 \), which boils down to \( x = \frac{1}{4} \). Therefore, the function intercepts the x-axis at \( \left( \frac{1}{4}, 0 \right) \).
Intercepts are essential for drafting accurate graphs and understanding how functions interact with coordinate systems.

Graph sketching combines all aspects—domain, range, asymptotes, and intercepts—to create a visual representation of the function. For \( f(x) = \frac{1}{x} - 4 \), begin by noting the intercepts and asymptotes.

• Plot the x-intercept at \( \left( \frac{1}{4}, 0 \right) \).
• Draw a vertical line at \( x = 0 \) for the vertical asymptote.
• Draw a horizontal line at \( y = -4 \) for the horizontal asymptote.

As \( x \) approaches either positive or negative infinity, the graph approaches but never touches \( y = -4 \). As \( x \) nears zero from the right, the function soars up to infinity, and from the left, it plummets down to negative infinity. Sketch these trends carefully, always approaching asymptotes yet never crossing them.
Creating a sketch of the graph aids significantly in understanding the function's long-term behavior and short-term variations, effectively tying numerous mathematical concepts together into a coherent, visual narrative.