If the domain of the function f(x) = √(x²-25)/(4-x²) + log10(x² + 2x - 15) is (-∞, α) U [β, ∞), then

Q: Wait, why do we reject the negative root here?

We do not reject the negative root; we reject values that make the argument negative. Remember, the square root of a negative number is not defined in the real number system. We are simply enforcing the rule that the radicand must be non-negative.

Q: Why is the final interval $(-\infty, -5)$ open at $-5$?

Excellent observation! While the square root condition allows $x = -5$ (because $\sqrt{0} = 0$), the logarithm condition $x^2 + 2x - 15 > 0$ requires the argument to be strictly greater than zero. Since $x = -5$ makes the log argument zero, it must be excluded from the final domain.

Q: Do we need to worry about the denominator $4 - x^2$ if the square root condition already restricts $x$?

Yes, absolutely! Never assume one condition covers another. Even if the square root restricts $x$ to values outside $(-5, 5)$, you must still explicitly check the denominator. In this specific problem, the excluded points $\pm 2$ fall within the region already excluded by the square root, but in other problems, they might fall inside the valid region. Always check every constraint.

Question

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

Options:

(A) 140

(B) 175

(C) 150

(D) 125

Sigma Percentile · Accepted Answer

### Analyzing the Function Constraints

To find the domain of the function $f(x) = \frac{\sqrt{x^2-25}}{4-x^2} + \log_{10}(x^2 + 2x - 15)$, we must satisfy three distinct mathematical conditions simultaneously.

The first checkpoint is the square root. For the expression to be defined in the real number system, the radicand must be non-negative:
$$x^2 - 25 \ge 0$$

Factoring this inequality, we get $(x - 5)(x + 5) \ge 0$. Using the wavy curve method, we determine the first interval:
$$x \in (-\infty, -5] \cup [5, \infty)$$

### Evaluating the Denominator and Logarithm

The second checkpoint involves the denominator of the fraction. A fraction is undefined if its denominator is zero, so we must ensure:
$$4 - x^2 
eq 0 \Rightarrow x^2 
eq 4 \Rightarrow x 
eq \pm 2$$

The third checkpoint is the logarithm. The argument of a logarithm must be strictly positive:
$$x^2 + 2x - 15 > 0$$

Factoring the quadratic, we obtain $(x + 5)(x - 3) > 0$. Applying the wavy curve method, we find the third interval:
$$x \in (-\infty, -5) \cup (3, \infty)$$

### Determining the Intersection

To find the final domain, we must find the intersection of all three sets:
1. $x \in (-\infty, -5] \cup [5, \infty)$
2. $x 
eq \pm 2$
3. $x \in (-\infty, -5) \cup (3, \infty)$

Observing the number line, the square root condition includes $-5$, but the logarithm condition excludes it. Thus, the intersection on the left side is $(-\infty, -5)$.

On the right side, the square root condition starts at $5$, which is already greater than $3$. Therefore, the intersection on the right side is $[5, \infty)$.

### Final Calculation

Combining these results, the domain of the function is $(-\infty, -5) \cup [5, \infty)$. Comparing this to the given form $(-\infty, \alpha) \cup [\beta, \infty)$, we identify:
$$\alpha = -5, \quad \beta = 5$$

We now calculate the final value:
$$\alpha^2 + \beta^3 = (-5)^2 + 5^3$$
$$\alpha^2 + \beta^3 = 25 + 125$$
$$\alpha^2 + \beta^3 = **150**$$

Animated Solution for Mathematics - Functions: If the domain of the function $f (x) = \frac{x ^{2} - 25}{( 4 - x ^{2} )} + lo g_{10} (x^{2} + 2 x - 15)$ is $(- \infty, α) \cup [β, \infty)$ , then $α^{2} + β^{3}$ is equal to :

Select Answer:

Visualized Solution

Analyzing the Function $f (x)$

The Three Constraints

Condition 1: Square Root

Solving Condition 1

Condition 2: Denominator

Solving Condition 2

Condition 3: Logarithm

Solving Condition 3

Intersection of All Conditions

The Final Domain

Finding $α$ and $β$

Calculating $α^{2} + β^{3}$

The Sigma Insight: Domain and Range of a Function

Analyzing the Function Constraints

Evaluating the Denominator and Logarithm

Determining the Intersection

Final Calculation

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