Find x-values if any at which the function f(x)=13x^2-15x-15 is not continuous. Which of the discontinuities are removable?
THANKS YOU!!
No, its not suppose to have a denominatory
all polynomials are continuous everywhere.
is this supposed to be a rational function?
well no its a parabola. so there are no discontinuius? How would I find the x-values if they are not.
Yes, it is a parabola which is a polynomial which is continuous.
Perhaps you have a typo and it is supposed to have a denominator?
well, since the question asks you to identify discontinuities "if any", the proper response is to say that there are none.
There is no value of x for which f(x) is not defined.
To determine if a function is continuous at a particular x-value, we need to check three conditions:
1. The function is defined at that x-value.
2. The limit of the function as x approaches that x-value exists.
3. The value of the function at that x-value matches the limit.
Now, let's apply these conditions to the function f(x) = 13x^2 - 15x - 15.
1. The function is defined for all real numbers because it is a polynomial.
2. To find the limits as x approaches different x-values, we need to calculate the left-hand limit and the right-hand limit separately. If the left-hand limit and the right-hand limit are the same, then the limit exists.
For x-values where the function is continuous, we do not need to calculate the limits. Discontinuities occur when the limits do not exist or when the function is not defined.
Now, let's find possible x-values at which the function is not continuous:
1. First, let's check if the function is defined at any specific x-value. In this case, since the function is a polynomial, it is defined for all real numbers.
2. Next, we need to find the limits as x approaches any specific x-value.
To find the limits, we need to find the left-hand limit and the right-hand limit separately. We will start by factoring the equation:
f(x) = 13x^2 - 15x - 15
We can factor the equation using quadratic factoring or apply the quadratic formula. However, in this case, it cannot be easily factored, so we will apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Using this formula, the roots (x-values) of the equation can be found by substituting the values of a, b, and c into the formula:
a = 13
b = -15
c = -15
Using the quadratic formula:
x = (-(-15) ± √((-15)^2 - 4 * 13 * -15)) / (2 * 13)
Simplifying:
x = (15 ± √(225 + 780)) / 26
x = (15 ± √1005) / 26
Now, we can calculate the specific x-values where the function may not be continuous. However, since √1005 ≈ 31.7, which is an irrational number, it means the function is continuous and defined for all real x-values.
Thus, there are no x-values at which the function f(x) = 13x^2 - 15x - 15 is not continuous. Therefore, there are no removable discontinuities in this function.