If the domain of the function f ( x ) = x 2 - 25 ( 4 - x 2 ) + log 10 ( x 2 + 2 x - 15 ) is ( - ? , ? ) ? [ ? , ? ) , then ? 2 + ? 3 is equal to: [2024]

Question

If the domain of the function $f (x) = \frac{\sqrt{x^{2} - 25}}{(4 - x^{2})} + \log_{10} (x^{2} + 2 x - 15)$ is $(- \infty, α) \cup [β, \infty),$ then $α^{2} + β^{3}$ is equal to: [2024]

Smart Achievers · Accepted Answer

(4)

$For domain, 4 - x^{2} \neq 0 \Rightarrow x \neq \pm 2$

$x^{2} - 25 \geq 0; x^{2} \geq 25 \Rightarrow x \in (- \infty, - 5] \cup [5, \infty)$

$Also, x^{2} + 2 x - 15 > 0 \Rightarrow (x + 5) (x - 3) > 0$

$\Rightarrow x \in (- \infty, - 5) \cup (3, \infty)$ $∴ D_{f} = (- \infty, - 5) \cup [5, \infty)$

$So, α = - 5 and β = 5$ $∴ α^{2} + β^{3} = 25 + 125 = 150$

Solution

Q.
If the domain of the function $f (x) = \frac{\sqrt{x^{2} - 25}}{(4 - x^{2})} + \log_{10} (x^{2} + 2 x - 15)$ is $(- \infty, α) \cup [β, \infty),$ then $α^{2} + β^{3}$ is equal to: [2024]

1 140

2 175

3 125

4 150

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