Final answer:
The discontinuities of the function x²+7x+1/x²+2x-15 are x=3 and x=-5.Option C is the correct answer.
Step-by-step explanation:
The discontinuities of a rational function occur when the denominator of the function equals zero. To find the value(s) of x that make the denominator zero, set x²+2x-15=0 and solve for x.
Factoring the quadratic equation, we get (x-3)(x+5)=0. Therefore, the discontinuities of the function are x=3 and x=-5.
Discontinuities in rational functions are closely tied to the points where the denominator becomes zero, causing the function to be undefined. In the case of the given rational function, the denominator is determined by the quadratic expression x² + 2x - 15.
To identify the x-values leading to discontinuities, this quadratic is factored as (x - 3)(x + 5) = 0.
By setting each factor to zero, we find the roots: x - 3 = 0 gives x = 3, and x + 5 = 0 yields x = -5. These are the critical points where the denominator equals zero, signifying the locations of potential discontinuities in the rational function.
Therefore, the function will be undefined at x = 3 and x = -5, emphasizing the importance of identifying these points to understand the behavior of the rational function and where it may not be continuous.