Are the following functions continuous?
{ x^2-1, if x>-1
f(x)= { 0, if x=-1
{ 4-x, if x<-1
A. yes
B. no
{ 2x+1, if x ≤-1
f(x)= { -x^2, if x>-1
A. yes
B. no
pls help
AAAaannndd the bot gets it wrong yet again!
#1: no, since f(-1)=0 but lim(x→-1-) = 5
#2: no, since f(-1)=1 but lim(x→-1-) = 1
oops. #2 No, because the right limit = 1, not the left
To determine if a function is continuous, we need to check if the function is defined for all values of x and if the limit of the function exists and is equal to the function value at each point.
For the first function, we have three pieces defined for different ranges of x:
1. For x > -1, the function is f(x) = x^2 - 1. This is a polynomial function, which is continuous for all real numbers. Therefore, this piece is continuous.
2. For x = -1, the function is f(-1) = 0. Since the function value for this specific point is defined, it is also continuous.
3. For x < -1, the function is f(x) = 4 - x. Similar to the first piece, this is a polynomial function and is continuous for all real numbers. Therefore, this piece is continuous as well.
Since all the pieces of the function are continuous for their respective ranges, we can conclude that the first function is continuous. The answer is A. yes.
Now, let's evaluate the second function in the same manner:
1. For x ≤ -1, the function is f(x) = 2x + 1. Again, this is a polynomial function and is continuous for all real numbers. Therefore, this piece is continuous.
2. For x > -1, the function is f(x) = -x^2. Once more, this is a polynomial function and is continuous for all real numbers. Therefore, this piece is continuous as well.
Since all the pieces of the function are continuous for their respective ranges, we can conclude that the second function is also continuous. The answer is A. yes.
Therefore, the answers to both questions are A. yes.
A. Yes
B. Yes