Which statement correctly describes the function?
f(x)= x+4/ x^2+7x+10
A. It is continuous.
B. It is discontinuous because there is a value a for which f(a) is not defined.
C. It is discontinuous because there is a value a such that lim x->a f(x) does not equal f(a).
D. It is discontinuous because there is a value a such that lim x->a f(x) does not exist.
B is good
it is discontinuous, because the denominator is zero at x = -2, -5
@ oobleck, so would the answer be B?
To determine if the given function f(x) = (x+4)/(x^2+7x+10) is continuous or discontinuous, we need to analyze its properties.
First, let's look at the denominator of the fraction, which is the expression x^2+7x+10. It is a quadratic equation. To determine if there are any values of x for which the denominator is equal to zero, we can find the roots of the quadratic equation by factoring or using the quadratic formula.
The roots of x^2+7x+10 = 0 can be found by factoring or applying the quadratic formula. The factored form is (x+5)(x+2) = 0, so the roots are x = -5 and x = -2.
Now, let's evaluate f(x) for these two values of x.
For x = -5:
f(-5) = (-5+4)/((-5)^2+7(-5)+10) = -1/(25-35+10) = -1/0
For x = -2:
f(-2) = (-2+4)/((-2)^2+7(-2)+10) = 2/(4-14+10) = 2/0
Notice that when we substitute these values into the function, we end up with a division by zero. Division by zero is undefined in mathematics.
Therefore, the correct statement describing the function f(x) is:
B. It is discontinuous because there is a value a for which f(a) is not defined.