Explain, using the theorems, why the function is continuous at every number in its domain.
F(x)= 2x^2-x-3 / x^2 +9
A) F(x) is a polynomial, so it is continuous at every number in its domain.
B) F(x) is a rational function, so it is continuous at every number in its domain.
C) F(x) is a composition of functions that are continuous for all real numbers, so it is continuous at every number in its domain.
D) F(x) is not continuous at every number in its domain.
E) none of these
State the domain. (Enter your answer using interval notation.)
The numerator is defined for every value of x you choose,
the denominator is always positive, so there is no danger of us dividing by zero, which would have caused a discontinuity.
so continuous for all values of x
B)
To determine whether the function F(x) = (2x^2 - x - 3) / (x^2 + 9) is continuous at every number in its domain, we can use the theorems related to continuous functions.
First, let's find the domain of the function. The denominator x^2 + 9 is a sum of squares, which is always nonnegative. Therefore, the denominator is always positive for any real value of x. We also note that the function is defined for all real numbers.
So, the domain of F(x) is (-∞, ∞).
Now, let's consider the theorems related to continuous functions:
1) A polynomial function is continuous for all real numbers. F(x) = 2x^2 - x - 3 is a polynomial function, so it is continuous at every number in its domain.
2) A rational function is continuous everywhere except at the points where the denominator is equal to zero. In this case, the denominator x^2 + 9 is never equal to zero for any value of x. Therefore, the function F(x) is continuous at every number in its domain.
3) A composition of functions that are continuous for all real numbers will also be continuous. In this case, F(x) can be considered as a composition of two functions: f(x) = 2x^2 - x - 3 and g(x) = x^2 + 9. Both f(x) and g(x) are continuous for all real numbers. Therefore, the composition F(x) is also continuous at every number in its domain.
Based on the above explanations, the correct answer is:
C) F(x) is a composition of functions that are continuous for all real numbers, so it is continuous at every number in its domain.
The domain is (-∞, ∞).
To determine whether the function is continuous at every number in its domain, we will use theorems related to the continuity of polynomials and rational functions.
First, let's identify the domain of the function. The function F(x) is a rational function, and the denominator of a rational function cannot be zero. Therefore, we need to find the values of x for which the denominator, x^2 + 9, is not zero. Since x^2 + 9 is always positive for any real value of x, the domain of F(x) is all real numbers, or (-∞, ∞) in interval notation.
Now, let's examine the theorems:
1. Theorem for Continuity of Polynomials:
A polynomial is continuous at every number in its domain. In this case, the numerator of F(x) is a polynomial, 2x^2 - x - 3, so by this theorem, F(x) is continuous at every number in its domain.
2. Theorem for Continuity of Rational Functions:
A rational function is continuous at every number in its domain, except where the denominator equals zero. In this case, the denominator of F(x) is x^2 + 9. However, as mentioned earlier, x^2 + 9 is always positive, making F(x) continuous at every number in its domain.
Based on these theorems, we can conclude that the correct answer is:
B) F(x) is a rational function, so it is continuous at every number in its domain.
The domain of F(x) is (-∞, ∞) in interval notation.