Final answer:
To find a hole in a function algebraically, factor the numerator and denominator, identify common factors that could be potential holes, set them equal to zero to find x-values, and plug x-values into the simplified function to find the y-values of the holes.
Step-by-step explanation:
Finding a hole in a function algebraically involves identifying points where the function is not defined due to an indeterminate form. To do this, first, look for common factors in the numerator and denominator of the function's rational expression. A hole occurs at the x-value that makes these factors equal to zero. Let's walk through this process step by step.
- Factor both the numerator and denominator of the function.
- Identify any common factors. These represent potential holes in the function.
- Set the common factors equal to zero to find the x-values of the holes.
- Plug these x-values into the simplified function (after removing the common factors) to find the corresponding y-values of the holes, emphasizing that the function is not defined at these points.
Let's consider an example function: f(x) = (x^2 - 4)/(x - 2). Factoring the numerator, we get f(x) = (x + 2)(x - 2)/(x - 2). The common factor is (x - 2). Setting x - 2 = 0, we find that x = 2 is where the hole is located. To find the y-coordinate, we simplify the function to f(x) = x + 2 and plug in x = 2 to get y = 4. Hence, there is a hole at the point (2, 4).
It's important to understand that a hole represents a point on the graph where the function does not exist, even though the function will approach this point from either side on the graph. In more complex functions, identifying holes may involve additional algebraic manipulation and applying concepts such as limits.