Q
MCQ
Calculus
If the domain of the function $f(x) = \sin^{-1} \left( \frac{5-x}{3+2x} \right) + \frac{1}{\log_e(10-x)}$ is $(-\infty, \alpha] \cup [\beta, \gamma) – \{\delta\}$, then $6(\alpha + \beta + \gamma + \delta)$ is equal to
1. Domain of Inverse Trigonometric Functions
The domain of a function $y = \sin^{-1}(x)$ is the interval $[-1, 1]$. When dealing with composite functions like $\sin^{-1}(g(x))$, we must ensure that the inner function $g(x)$ satisfies the inequality $-1 \le g(x) \le 1$. This often leads to rational inequalities that are solved using the method of intervals (wavy curve method). It is also crucial to ensure that $g(x)$ itself is defined, meaning any denominators within $g(x)$ cannot be zero.
2. Logarithmic Function Constraints
For a logarithmic function $\log_b(a)$ to be real-valued, two primary conditions must be met: the argument $a$ must be strictly positive ($a > 0$), and the base $b$ must be positive and not equal to 1 ($b > 0, b \neq 1$). In this problem, the natural log $\log_e(10-x)$ requires $10-x > 0$. Furthermore, because the log term is in the denominator, the entire expression $\log_e(10-x)$ must not equal zero, which imposes the additional restriction that $10-x \neq 1$.
3. Intersection of Domains
The domain of the sum or difference of two functions, $f(x) = g(x) + h(x)$, is the intersection of the domains of $g(x)$ and $h(x)$. If $D_g$ is the set of values for which $g$ is defined and $D_h$ is the set for $h$, then $D_f = D_g \cap D_h$. This ensures that every component of the function is simultaneously valid for a given $x$. This principle extends to any number of algebraic combinations of functions.
4. Solving Rational Inequalities
To solve inequalities of the form $\frac{P(x)}{Q(x)} \ge 0$, one should find the roots of both numerator and denominator. These roots divide the number line into intervals. By testing a point in each interval or using the signs of the factors (wavy curve), we determine where the expression is positive or negative. Note that while numerator roots can be included for “$\ge$” or “$\le$” signs, denominator roots must always be excluded as they lead to undefined values.
Q
Why is the domain of sin⁻¹(x) limited to [-1, 1]?
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Because sin(y) = x only has solutions for x in [-1, 1]. The inverse function is only defined within the range of the original function.
Q
What if the log base was not ‘e’?
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The argument condition (10-x > 0) remains the same regardless of the base, as long as the base is positive and not 1.
Q
How do we handle the -3/2 point?
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x = -3/2 makes the denominator of (5-x)/(3+2x) zero, so it must be excluded from the domain using open parentheses.
Q
Why did we subtract {9}?
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Because at x = 9, log(10-9) = log(1) = 0. Since the log is in the denominator, x=9 makes the function undefined.
Q
What is the wavy curve method?
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It is a technique to solve polynomial and rational inequalities by plotting roots on a number line and checking signs of intervals.
Q
Can the domain be empty?
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Yes, if the intersection of the conditions for different parts of the function yields no common x-values.
Q
How to handle 1/log(f(x))?
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You need f(x) > 0 (for log to exist) AND f(x) ≠ 1 (so log is not 0 in the denominator).
Q
Why is α = -8 and not 10?
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Based on the structure (-∞, α] ∪ [β, γ), α represents the upper bound of the first closed interval, which is -8.
Q
Is sin⁻¹(u) defined for u > 1?
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No, the inverse sine function is not defined for values outside the range [-1, 1].
It refers to the natural logarithm (ln), where the base is Euler’s number e (≈ 2.718).