State whether the following functions are continuous or discontinuous. if it is discontinuous, state the value of x which creates discontinuously.
a) f(x) = x sin x
b) f(x) = (x^2+x-12)/(x-3)
c) f(x) = √x + ((1)/(x-1))
d) f(x) =
{x-5 x <= 1
{7 x > 1
part d, it is only one curly bracket.
(a) since both x and sinx are continuous, so is x sinx
(b) division by zero is undefined, so there is a discontinuity at x=3
(c) same problem as (b)
(d) f(1) = 1-5 = -4
limit(x→1+) = 7*1 = 7
since the two limits are different, f(x) is discontinuous at x=1
To determine whether a function is continuous or discontinuous, we need to check three conditions:
1. The function must be defined at the given point.
2. The limit of the function must exist at that point.
3. The limit must be equal to the value of the function at that point.
Now, let's analyze each function to determine whether it is continuous or discontinuous.
a) f(x) = x sin(x)
This function is continuous for all values of x. There are no specific values of x that would make it discontinuous.
b) f(x) = (x^2+x-12)/(x-3)
To determine if this function is continuous or discontinuous at x = 3, we need to check the limit as x approaches 3. By substituting x = 3, we get an undefined expression because of division by zero. Therefore, the function is discontinuous at x = 3.
c) f(x) = √x + (1)/(x-1)
At x = 1, the function becomes discontinuous because the denominator (x-1) becomes zero. Additionally, the term √x is not defined for negative values. Therefore, the function is discontinuous at x = 1.
d) f(x) = {x-5, x ≤ 1, 7, x > 1}
This function is defined using different formulas for different intervals. For x ≤ 1, the function is continuous as it is a linear function. For x > 1, the function is also continuous as it is a constant function. However, at x = 1, the two pieces of the function do not agree. The limit from the left side (approaching x = 1) is 1 - 5 = -4, while the limit from the right side (approaching x = 1) is 7. Since the left and right limits don't agree, the function is discontinuous at x = 1.
Therefore:
a) Continuous for all values of x.
b) Discontinuous at x = 3.
c) Discontinuous at x = 1.
d) Discontinuous at x = 1.