Final answer:
The graph of f(x) = |x| is a V-shaped curve that opens upwards. g(x) = ||2x|| and h(x) = ||3x|| have similar shapes, but with increased steepness due to the multiplication factor. For k(x) = ||nx||, the graph becomes steeper as n increases.
Step-by-step explanation:
To sketch the graph of a function, we need to understand its behavior for different values of x. Let's first consider the function f(x) = ||x||. The absolute value of x, denoted as |x|, represents the distance of x from 0 on the number line. So, when we take the absolute value of |x|, it always gives a positive value. So, for f(x) = |x|, the graph will be a V-shaped curve that opens upwards. For g(x) = ||2x||, the graph will be similar to f(x), but the steepness will be doubled because of the factor of 2 in front of x. Lastly, for h(x) = ||3x||, the steepness will be tripled compared to f(x) because of the factor of 3 in front of x. Now, for the function k(x) = ||nx||, the graph will have a steeper V-shaped curve where the steepness increases as the value of n increases. So, as n is a positive integer, the graph of k(x) = ||nx|| will become gradually steeper as n increases.