f(x)= xsquared - 2x
Determine whether y= f(IxI) is continous at x= 0 justify your answer
1. determine if f(|x|) exists at x=0.
2. If it exists, calculate the limites of the following as x->0
a. f(|x|)
b. f(|x+|)
c. f(|x-|)
If all three have identical values, f(x) is continuous at f(|0|).
See:
http://img683.imageshack.us/img683/654/1285202904.png
To determine whether the function y = f(|x|) = |x|^2 - 2|x| is continuous at x = 0, we need to check the following conditions:
1. The function f(|x|) is defined at x = 0.
2. The limit of f(|x|) as x approaches 0 exists.
3. The value of f(|x|) at x = 0 is equal to the limit.
Let's first substitute x=0 into the function y = f(|x|):
f(|0|) = |0|^2 - 2|0| = 0 - 0 = 0
Therefore, the function f(|x|) is defined at x = 0.
Now, let's find the limit of f(|x|) as x approaches 0. To find the limit, we need to consider the left and right limits separately.
For x < 0:
f(|x|) = |-x|^2 - 2|-x| = x^2 - 2x
Taking the limit as x approaches 0 from the left, we have:
lim (x → 0-) (x^2 - 2x) = 0
For x > 0:
f(|x|) = |x|^2 - 2|x| = x^2 - 2x
Taking the limit as x approaches 0 from the right, we have:
lim (x → 0+) (x^2 - 2x) = 0
Since the left and right limits of f(|x|) as x approaches 0 both equal 0, the limit of f(|x|) as x approaches 0 exists and is equal to 0.
Lastly, we compare the value of f(|x|) at x = 0 with the limit:
f(|0|) = 0
lim (x → 0) (x^2 - 2x) = 0
The value of f(|x|) at x = 0 is equal to the limit. Therefore, y = f(|x|) is continuous at x = 0.
In summary, we concluded that y = f(|x|) is continuous at x = 0 because the function is defined at x = 0, the limit of f(|x|) as x approaches 0 exists, and the value of f(|x|) at x = 0 is equal to the limit.