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Understanding the x-intercept of a function is fundamental when sketching graphs. The x-intercept is where the graph of the function crosses the x-axis. It's found by setting the function equal to zero and solving for x. For instance, with the function f(x) = (x-4)/x^2, we set (x-4)/x^2 = 0 and solve, which gives us x=4. This is the point where the graph touches the x-axis. It's important for students not to forget that there can be multiple x-intercepts or none, depending on the function.

A vertical asymptote represents a value that the function approaches but never actually reaches. It maps out a line on the graph that the curve gets infinitely close to but does not cross. To find a vertical asymptote, we look for values that make the denominator of a fraction undefined, as the function cannot exist at these points. For the example f(x) = (x-4)/x^2, setting the denominator equal to zero with x^2 = 0 gives us x = 0. Hence, the line x = 0 is our vertical asymptote. These are crucial for shaping the graph and understanding the function's behavior.

The horizontal asymptote of a function is a horizontal line that the graph approaches as x moves towards infinity or -infinity. To determine this for a rational function, we compare the degrees of the numerator and denominator. If the degree of the numerator is less, like in f(x) = (x-4)/x^2 where it's 1 compared to 2, the horizontal asymptote is y=0. This line helps us predict the end behavior of the graph. A function may approach this asymptote from above or below, and students should note that a function with a higher degree in the numerator could have an oblique asymptote instead.

The domain of a function is the set of all possible input values (usually x-values), which make the function work without causing contradictions or undefined terms. For our function f(x) = (x-4)/x^2, the domain is all real numbers except for x=0, as at zero the denominator becomes undefined and thus the function does not exist. This is expressed as x eq 0.

The range, on the other hand, is all possible outputs (usually y-values) that the function can produce. In our case, the function can take any y-value except y=0, since this is the horizontal asymptote, which the function will approach but can never reach. Again, we state this as y eq 0. Understanding both domain and range helps students anticipate which x and y values are acceptable when graphing and interpreting functions.