Find the values of x for which each function is continous:
f(x) = x^3 -2x^2+x -1
look at your expression.
Is there any value of x which when substituted does not give you a real number?
To find the values of x for which the function f(x) = x^3 - 2x^2 + x - 1 is continuous, we need to check the three conditions for continuity:
1. The function must be defined at the given x-values.
2. The left-hand limit and the right-hand limit must exist at each x-value.
3. The function value must be equal to the limits at each x-value.
Let's go through each condition one by one:
1. The function f(x) = x^3 - 2x^2 + x - 1 is a polynomial, which means it is defined for all real numbers.
2. To check the limits, we need to evaluate the function from the left and the right side of each x-value. If the left-hand limit is equal to the right-hand limit, then the limit exists.
Let's find the limits:
For x → a⁻ (left-hand limit):
lim (x → a⁻) f(x) = lim (x → a⁻) (x^3 - 2x^2 + x - 1)
For x → a⁺ (right-hand limit):
lim (x → a⁺) f(x) = lim (x → a⁺) (x^3 - 2x^2 + x - 1)
3. Finally, we compare the function value at x = a to the limits we found. If they are equal, then the function is continuous at x = a.
Now, let's solve for x to find the values for which the function is continuous:
Since the given function is a polynomial with no restrictions or removable discontinuities, it is continuous for all real numbers. In other words, there are no specific values of x for which the function is not continuous.
Therefore, the function f(x) = x^3 - 2x^2 + x - 1 is continuous for all values of x.