Identify a rational function whose graph is a horizontal line except for two holes.
Graph function.
To identify a rational function with a horizontal line graph and two holes, we can start by understanding the characteristics of a horizontal line graph. A horizontal line is represented by an equation in the form of y = constant. This means that the function will have the same output (y-value) regardless of the input (x-value).
To introduce two holes in the graph, we need to have two values of x for which the function is undefined, creating gaps or holes in the graph. This can be achieved by introducing a rational function with factors in both the numerator and denominator that cancel out at those specific x-values.
An example of such a rational function is:
f(x) = (x^2 - 4) / (x + 2)
This function has a horizontal line graph since the numerator (x^2 - 4) does not depend on x; it is a constant for any value of x. The denominator (x + 2) cancels out the (x - 2) term in the numerator, creating two holes in the graph at x = -2 and x = 2.
To graph this function, you can follow these steps:
1. Determine the x-values for the holes: In this example, the function has holes at x = -2 and x = 2.
2. Plot the holes: Mark the points (-2, f(-2)) and (2, f(2)) on the graph. The y-values at these points can be found by substituting the x-values into the function.
3. Draw a dotted line: Connect the points on the graph with a dotted line to represent the function's horizontal behavior.
The resulting graph will be a horizontal line with two holes at the specified x-values.