The domain of definition of the function f(x) | vedclass

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Question

(C) For the function $f(x)$ to be defined,we must satisfy three conditions: $1$. The binomial coefficients and permutations must have non-negative int

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$\{2, 3\}$

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(C) For the function $f(x)$ to be defined,we must satisfy three conditions: $1$. The binomial coefficients and permutations must have non-negative integer parameters: For $^{16-x}C_{2x-1}$,we need $16-x \ge 2x-1 \ge 0$,which implies $17 \ge 3x$ $(x \le 17/3)$ and $2x \ge 1$ $(x \ge 1/2)$. For $^{20-3x}P_{2x-5}$,we need $20-3x \ge 2x-5 \ge 0$,which implies $25 \ge 5x$ $(x \le 5)$ and $2x \ge 5$ $(x \ge 2.5)$. Combining these,we get $2.5 \le x \le 5$. $2$. The base of the logarithm $[x + \frac{1}{x}]$ must be greater than $0$ and not equal to $1$. For $x \in [2.5, 5]$,$x + \frac{1}{x}$ ranges from $2.5 + 0.4 = 2.9$ to $5 + 0.2 = 5.2$. Thus,$[x + \frac{1}{x}]$ can take values $2, 3, 4, 5$. Since $[x + \frac{1}{x}] 
eq 1$ is satisfied (as the minimum value is $2$),the condition is satisfied for all $x \in [2.5, 5]$. $3$. The argument of the logarithm $|x^2 - x - 6|$ must be greater than $0$. $|x^2 - x - 6| > 0 \implies x^2 - x - 6 
eq 0 \implies (x-3)(x+2) 
eq 0$. So $x 
eq 3$ and $x 
eq -2$. Given $x \in [2.5, 5]$,we must exclude $x=3$. Checking integer values in the range $[2.5, 5]$: $x$ can be $3, 4, 5$. However,the binomial coefficients require $x$ to be such that $2x-1$ and $2x-5$ are integers,implying $2x$ must be an integer. Testing $x=3$: $f(3) = \log_{[3+1/3]} |9-3-6| + ^{13}C_5 + ^{11}P_1 = \log_3(0) + \dots$,which is undefined. Testing $x=4$: $f(4) = \log_{[4+0.25]} |16-4-6| + ^{12}C_7 + ^{8}P_3 = \log_4(6) + 792 + 336$,which is defined. Testing $x=5$: $f(5) = \log_{[5+0.2]} |25-5-6| + ^{11}C_9 + ^{5}P_5 = \log_5(14) + 55 + 120$,which is defined. Since the domain must be discrete due to the nature of $nCr$ and $nPr$ with integer constraints,the values are $\{4, 5\}$. Re-evaluating the options,there appears to be a discrepancy. Given the standard constraints,the set is $\{4, 5\}$.

The domain of definition of the function $f(x) = \log_{[x + \frac{1}{x}]} |x^2 - x - 6| + ^{16-x}C_{2x-1} + ^{20-3x}P_{2x-5}$ is,where $[x]$ denotes the greatest integer function.

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