Find the Domain y=(t^3+t)/(2t^3+1)

Problem

y=(t^3+t)/(2*t^3+1)

Solution

1. Identify the condition for the domain of a rational function, which requires that the denominator cannot be equal to zero.

2*t^3+1≠0

2. Isolate the variable term by subtracting 1 from both sides of the inequality.

2*t^3≠−1

3. Divide both sides by 2 to solve for the cubed variable.

t^3≠−1/2

4. Take the cube root of both sides to find the excluded value for t

t≠√(3,−1/2)

5. Simplify the radical expression by extracting the negative sign, as the cube root of a negative number is real.

t≠−1/√(3,2)

6. State the domain in interval notation, excluding the value where the function is undefined.

(−∞,−1/√(3,2))∪(−1/√(3,2),∞)

Final Answer

Domain: *{[t∈ℝ,t≠−1/√(3,2)]}

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