If the domain of the function
f(x)=√(x²-25)/(4-x²)+log ₁₀(x²+2 x-15) is (-∞, α) ∪[β, ∞), then α²+β³ is equal to :

Solution

Maths sir · 3 steps

Write down every condition the function needs, solve each one, then keep only the x that pass all of them.

1. List the conditions. The square root needs a non-negative inside, the denominator must not be zero, and the log needs a positive argument. The domain is where all three hold together.

   Part of f(x)	Condition	Solution set
   √(x²-25)	x²-25 ≥ 0	x ≤ -5 or x ≥ 5
   1/(4-x²)	4-x² ≠ 0	x ≠ ± 2
   log₁₀(x²+2x-15)	(x+5)(x-3) > 0	x < -5 or x > 3

2. Intersect them. On the left, x ≤ -5 and x < -5 overlap in x < -5, because x = -5 makes the log argument zero. On the right, x ≥ 5 and x > 3 overlap in x ≥ 5. The points ± 2 are already outside both pieces.

   Domain = (-∞, -5) ∪ [5, ∞)

3. So. α = -5 and β = 5, so α² + β³ = 25 + 125 = 150, which is option (d).

The idea to keep. For a domain, take each piece separately, solve its condition, then intersect. Watch the endpoints: a square root allows zero, a log does not.

Answer from the NTA final answer key. Solution by PrepWiser.