the vector function r(t)= <(t-1)/(t+1), sin(t) , ln(25-t^2)
find the domain of r and answer in interval notation.
will domain have any brackets?
The domain of a vector function is the set of all values of t for which the function is defined.
For this function, we need to consider the domain of each component separately.
1. The first component, (t-1)/(t+1), is defined for all real values of t except t=-1. Therefore, the domain of the first component is (-∞,-1) U (-1, ∞).
2. The second component, sin(t), is defined for all real values of t. Therefore, the domain of the second component is (-∞,∞).
3. The third component, ln(25-t^2), is defined only for 5≤t≤-5. This is because the argument of the natural logarithm must be positive. Therefore, we need to solve the inequality 25-t^2>0:
t^2<25
|t|<5
t in (-5, 5)
Therefore, the domain of the third component is (-5, 5).
To determine the domain of the entire vector function, we need to find the intersection of the domains of each component.
The intersection of (-∞,-1) U (-1,∞) and (-5,5) is (-5,-1) U (-1,5). Therefore, the domain of the vector function r(t) is (-5,-1) U (-1,5).
There are parentheses used in the interval notation since the endpoints are not included in the domain.
The domain of r is: (-5,-1) U (-1,5)