Alright, let's solve this step by step!
We are given the absolute value function f(x) = |x| - 6 and it is asked to find the solution.
We first need to understand what an absolute value function does. With absolute value, we are always looking at the distance from 0, so the result of absolute value is always nonnegative.
In this case, to find the values for which this function equals 0, we set f(x) equal to 0, and proceed as follows:
0 = |x| - 6
The absolute value function will look like a 'V' shape. The '-6' in the function represents a vertical shift down by 6 units, affecting where the vertex point is.
Now, we rearrange the equation to find |x|, so we add 6 to both sides of the equation and get:
6 = |x|
This tells us that the distance from x to 0 is 6. Since this distance could be either in the positive or negative direction, there are two solutions to this equation: x = -6 and x = 6. Both values when placed into the absolute value function would equal to 6.
That's the solution to the absolute value function! However, we are also asked to rewrite the function f(x)=|x|-6.
Such a function can be defined in a piecewise manner, as follows:
- For all values of x that are greater than or equal to 0, function f(x) takes a form of x - 6. Hence when x >= 0, f(x) = x - 6.
- For all values of x that are less than 0, function f(x) takes a form of -x - 6. Hence when x < 0, f(x) = -x - 6.
So, to summarize:
The absolute value function f(x) = |x| - 6 has solutions at x=6 and x=-6. The rewritten piecewise function is f(x) = x-6 when x>=0, and f(x) = -x-6 when x<0.
And that's it! We have solved the problem.