How do you graph #f(x)=abs((x)-1)-(x)#?

Question

Avery Alexander · Accepted Answer

See explanation

#color(red)("Identify the critical points and analyse the behaviour at the point and either side")#

#color(blue)("Consider the case where "x<0)#

Then #|x-1|# is the same as positive #x+1#
now subtract the #-(x)# remembering that #x# is negative.
The net result is #y=positive(2x)+1#
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#color(blue)("I will let you figure out "0 < x < 1)#
The graph shows what you should end up with.
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#color(blue)("Consider the case where "x=1)#

#|x-1|=0" so "|x-1|-1=-1#
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#color(blue)("Consider the case where "x>1)#

#|x-1| " is the same as "x-1#

Put it all together

#|x-1|-x" "->" "x-1-x=1#
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Kennedy Adams · Answer

undefined To graph the function f(x) = |x - 1| - x, you can follow these steps:

1. Identify critical points where the function changes behavior. In this case, the critical point is x = 1.
2. Determine the behavior of the function around the critical point by considering intervals on either side of x = 1.
3. Plug in test points within each interval to determine the sign of f(x) in that interval.
4. Plot the critical point and the test points on the graph.
5. Connect the points to create the graph of the function.

Here's a brief description of the graph's behavior:

- For x < 1, f(x) = -(x - 1) - x = -2x + 1. So, the graph is a line with a slope of -2 and y-intercept of 1.
- For x > 1, f(x) = (x - 1) - x = -1. So, the graph is a horizontal line at y = -1.

Combining these, the graph will have a corner at the point (1, -1), and on the left side of 1, it will slope downward with a slope of -2, and on the right side of 1, it will be a horizontal line at y = -1.