Given that function
f(x) = 9, x<4
2x+1, 4≤x<12
7x-4, x≥12
Is the function continuous
a) as x tends to 4?
b) as x tends to 12?
look continuous to me
notice you have the straight line y = 9 as long as x < 4
at x ≥ 4 the line y = 2x+1 takes over all the way to x < 12
As soon as x reaches 12, the baton is passed to y = 7x - 4
at x = 12
2x+1 = 25
7x-4 = 80
To determine if a function is continuous at a specific point, we need to check if the limit of the function exists as x approaches that point and if the value of the function at that point is equal to the limit.
a) To check if the function is continuous as x tends to 4, we need to evaluate the limit of the function as x approaches 4 from both the left and the right sides.
From the left side (x < 4):
lim (x→4-) f(x) = lim (x→4-) 9 = 9
From the right side (x ≥ 4):
lim (x→4+) f(x) = lim (x→4+) (2x+1) = 2(4) + 1 = 9
The limit of the function from both sides is equal to 9. Also, f(4) = 9. Therefore, the function is continuous at x = 4.
b) To check if the function is continuous as x tends to 12, we need to evaluate the limit of the function as x approaches 12 from both the left and the right sides.
From the left side (x < 12):
lim (x→12-) f(x) = lim (x→12-) (2x+1) = 2(12) + 1 = 25
From the right side (x ≥ 12):
lim (x→12+) f(x) = lim (x→12+) (7x-4) = 7(12) - 4 = 80
The limit of the function from the left side is 25, and from the right side is 80. Since these limits are not equal, the function is not continuous at x = 12.