Find the domain of the vector-valued functions: (a) r→(t)=⟨t−1,(t+1)−1,sin−1(t)⟩ (b) r→(t)=⟨8−t3,In(t),et⟩

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Find the domain of the vector-valued functions:  (a) r→(t)=⟨t−1,(t+1)−1,sin−1(t)⟩  (b) r→(t)=⟨8−t3,In(t),et⟩

Barbara Meeker · Accepted Answer

Step 1 To Determine: find the domain of the vector-valued function Given: we have given two functions a) r→(t)=⟨t−1,(t+1)−1,sin−1(t)⟩ b) r→(t)=⟨8−t3,ln⁡(t),et⟩ Explanation: Part a) we have a function r→(t)=⟨t−1,(t+1)−1,sin−1(t)⟩ Here we see the domain of each individual separately Domain of t−1=1t is t&amp;gt;0 that is R−0 or (−∞,0)∪(0,∞)…(1) Domain of  (t+1)−1=1t+1 is t+1≠0⇒t≠−1 that is R−−1, or (−∞,−1)∪(−1,∞)…(2) Domain of sin−1⁡t is [−1,1]…(3) so the domain of the whole funcion will be the common region of all the three. here we see that the function is not defined at t=O and t=−1 from the above we conclude that the domainis (−1,0)∪(0,1] Step 2 Part b) we havea function r→(t)=⟨8−t3,ln⁡(t),et⟩ Domain of 8−t3 is 8−t3≥0 ⇒−t3≥−8 ⇒−t3≤8 ⇒−t3≤2 or (−∞,2]…(1) Domain of ln⁡(t) is t&amp;gt;0 or (0,∞)…(2) Domain of et is t≥0 or [0,∞)…(3) from the above three the common region will be the domain of the function . Domain is (0,2]

Find the domain of the vector-valued functions: (a) \vec{r}(t)=<< t^{-1},(t+1)^{-1},\sin^{-1}(t)>> (b)

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