### Instruction: What is the range of the function $$F(x) = |x+1|+|x-5|~?$$Express your answer in interval notation.### Response: To find the range of a function, I need to consider what values it can take for different inputs of x.
I notice that the function has two absolute value expressions, which means it is composed of four linear pieces.
To identify the pieces, I can look at the critical points where the absolute value expressions change sign.
These are x = -1 and x = 5.
For x < -1, both expressions are negative, so I have F(x) = -x-1-x+5 = -2x+4.
For -1 <= x < 5, the first expression is positive and the second is negative, so I have F(x) = x+1-x+5 = 6.
For x >= 5, both expressions are positive, so I have F(x) = x+1+x-5 = 2x-4.
I can graph these pieces on a coordinate plane to see how they look.
I see that the function has a minimum value of 6 at x = -1 and x = 5, and it increases without bound as x goes to the left or to the right.
Therefore, the range of the function is all real numbers greater than or equal to 6.
In interval notation, this is [6, infinity).