2. find the domains of the vector valued function\n\n$f(t)=langle t^{3},\frac{t}{t + 3}\rangle$
2. find the domains of the vector valued function\n\n$f(t)=langle t^{3},\frac{t}{t + 3}\rangle$
Answer
Answer:
$(-\infty, - 3)\cup(-3,\infty)$
Explanation:
Step1: Analyze each component
The first - component $t^3$ has domain $(-\infty,\infty)$ as it is a polynomial. The second - component is $\frac{t}{t + 3}$, and for a rational function, the denominator cannot be zero.
Step2: Set denominator not equal to zero
Set $t+3\neq0$. Solving for $t$, we get $t\neq - 3$.
Step3: Determine the domain of the vector - valued function
The domain of the vector - valued function is the intersection of the domains of its components. Since the domain of $t^3$ is all real numbers and the domain of $\frac{t}{t + 3}$ is all real numbers except $t=-3$, the domain of $f(t)$ is $(-\infty, - 3)\cup(-3,\infty)$.