Question

Apollo graphed the following function using Desmos, as shown below, and elaimed that the function f(x)=(x²+x-6)/(x²--9x+14) is only discontinuous at the point x=7. Is Apollo correct?

Answer 1

Apollo is partially correct; while f(x) is indeed discontinuous at x=7, there is also a hole at x=2 due to the factor (x - 2) being canceled in the simplification process. There are discontinuities at both x=2 and x=7.

Step-by-step explanation:

The question asks about the continuity of the function f(x) = (x² + x - 6) / (x² - 9x + 14). To determine if Apollo is correct in claiming the function is only discontinuous at x=7, we need to factor both the numerator and the denominator and simplify if possible.

First, the numerator can be factored as (x + 3)(x - 2), and the denominator factors into (x - 7)(x - 2). After canceling out the (x - 2) terms, you're left with f(x) = (x + 3) / (x - 7). Apollo is partially correct; the function is discontinuous at x=7 due to the denominator becoming zero, which results in undefined behavior. However, we must also consider possible discontinuities from the original denominator before simplification. Since the original factor (x - 2) was canceled out in the simplification, it implies there might be a hole in the graph at x=2. Therefore, the function is discontinuous at both x=2 and x=7.

Considering if we were to graph f(x) for 0 ≤ x ≤ 20, we would observe the hole at x=2 and the vertical asymptote at x=7, showing the discontinuities of the function within the given interval.