Final answer:
To find points with integer coordinates for the function f(x) = -log4(x+6)+1, we need values of x where x+6 is a power of 4. One such point is (-2, 0), where x+6 equals 4, and a second point is (-5, 1), where x+6 equals 1, which is 4 to the power of 0.
Step-by-step explanation:
The question asks to find any two points with integer coordinates for the function f(x) = -log₄(x+6)+1. To find such points, we would seek integer values of x where x+6 is a power of 4, as logarithms yield integer results for inputs that are exact powers of their base.
Let's consider x=2. Plugging in, we get f(2) = -log₄(2+6) + 1 = -log₄(8) + 1. Since 8 is not a power of 4, this will not give an integer result for f(2). However, if x=-4, we have f(-4) = -log₄(-4+6) + 1 = -log₄(2) + 1. Since 2 is 4 to the power of -1/2, we get an integer value of f(-4) = 1 - (-1/2) = 3/2, which is not an integer.
Let's try another value: x=-2. Then f(-2) = -log₄(-2+6) + 1 = -log₄(4) + 1. In this case, since 4 is 4 to the power of 1, we get f(-2) = 1 - 1 = 0, which is an integer.
So one point with integer coordinates is (-2, 0). Continuing this process, if x=-6, then f(-6) = -log₄(-6+6)+1 = -log₄(0) + 1, but logarithm of zero is undefined, so no integer coordinate exists for x=-6.
For the second point, consider x=4. We have f(4) = -log₄(4+6) + 1 = -log₄(10) + 1, which also does not yield an integer since 10 is not a power of 4.
We need to find an x such that x+6 is a power of 4. A logical choice is x=-5, so that x+6=1, which is 4 to the power of 0. Then, f(-5) = -log₄(1) + 1 = 0 + 1 = 1, giving us a second point with integer coordinates (-5, 1).