Answer:
To graph the function f(x)=[x+6], we need to understand what the brackets mean. The brackets are called the floor function, which rounds down any real number to the nearest integer. For example, [3.2]=3, [-1.7]=-2, and [0]=01. The floor function creates a piecewise constant function, which means that the function has different constant values for different intervals of x2. To graph f(x)=[x+6], we can follow these steps:
Find the intervals of x where f(x) is constant. This happens when x is between two consecutive integers, such as 0<x<1, 1<x<2, 2<x<3, and so on.
For each interval, find the value of f(x) by adding 6 to the lower endpoint of the interval and rounding down. For example, for 0<x<1, f(x)=[0+6]=6; for 1<x<2, f(x)=[1+6]=7; for 2<x<3, f(x)=[2+6]=8, and so on.
Plot the points (x,f(x)) for each interval, and connect them with horizontal line segments. The graph will look like a staircase that goes up by one unit every time x increases by one unit.
The domain of f(x) is the set of all possible values of x that make sense for the function. Since x can be any real number, the domain of f(x) is all real numbers, or (-∞,∞). The range of f(x) is the set of all possible values of f(x) that are produced by the function. Since f(x) is always an integer that is greater than or equal to 6, the range of f(x) is all integers that are greater than or equal to 6, or [6,∞).