r(t)= < lnsqrt(t+2) , 4t , t/(9-t^2) >
find domain of r and answer in interval notation
The domain of r(t) depends on the domain of each component function.
The first component function is ln(sqrt(t+2)). For the natural logarithmic function ln(x), the argument (x) must be greater than zero. Therefore, for sqrt(t+2) to be inside the domain of ln, t+2 must be greater than zero, which means t must be greater than -2.
The second component function is 4t, which has a domain of all real numbers.
The third component function is t/(9-t^2). The denominator cannot be zero or negative, so we must solve the inequality 9-t^2 > 0. This inequality can be factored as (3+t)(3-t) > 0, which means t must be less than -3 or greater than 3. However, we also need to exclude any values of t that make the denominator zero, so we must exclude the values t = -3 and t = 3.
Therefore, the domain of r(t) is (-2, -3) U (-3, 3) U (3, infinity) in interval notation.