Final answer:
The function F(x) is continuous at all points except where the denominator is zero, which are x = 5 and x = 9, making it continuous at x = 15 and x = 45.
Step-by-step explanation:
The student is asking for the continuity of the rational function F(x) = \frac{x^2-7x-10}{x^2-14x+45}. for a function to be continuous at a point x = a, the function must be defined at that point, which means the denominator should not be equal to zero. first we should factor the denominator. the denominator can be factored as (x - 9)(x - 5) which means that the function will have discontinuities at x = 9 and x = 5 as the function becomes undefined at these points. Next, factoring the numerator we get (x - 10)(x + 1). Now, we can see that the function has discontinuities at x = 5 and x = 9, but it is continuous at other values of x given in the options, namely x = 15 and x = 45.
Answer: b) x = 9, c) x = 15, d) x = 45