Vincen wrote: ↑Wed Sep 30, 2020 6:54 am
For which of the following functions does \(f(x) = f\left(\dfrac1{x}\right),\) given that \(x \ne - 2, - 1, 0,\) or \(1?\)
A. \(f(x)=\left|\dfrac{x+1}{x}\right|\)
B. \(f(x)=\left|\dfrac{x+1}{x-1}\right|\)
C. \(f(x)=\left|\dfrac{x-1}{x}\right|\)
D. \(f(x)=\left|\dfrac{x}{x+1}\right|\)
E. \(f(x)=\left|\dfrac{x+1}{x+2}\right|\)
Answer:
B
Solution:
The easiest way to solve the problem (without solving the problem algebraically) is to use a value for x (other than -2, -1, 0, and 1). So, let’s use x = 2.
A) f(2) = |(2 + 1)/2| = 3/2 f(1/2) = |(1/2 + 1)/(1/2)| = (3/2)/(1/2) = 3
Since f(2) ≠ f(1/2), A is not the correct answer.
B) f(2) = |(2 + 1)/(2- 1)| = 3/1 = 3 f(1/2) = |(1/2 + 1)/(1/2 - 1)| = |(3/2)/(-1/2)| = |-3| = 3
Since f(2) = f(1/2), B could be the correct answer so long as we can show that none of the three remaining choices can be the correct answer.
C) f(2) = |(2 - 1)/2| = 1/2 f(1/2) = |(1/2 - 1)/(1/2)| = |(-1/2)/(1/2)| = |-1| = 1
Since f(2) ≠ f(1/2), C is not the correct answer.
D) f(2) = |2/(2 + 1)| = 2/3 f(1/2) = |(1/2)/(1/2 + 1)| = (1/2)/(3/2) = 1/3
Since f(2) ≠ f(1/2), D is not the correct answer.
E) f(2) = |(2 + 1)/(2 + 2)| = 3/4 f(1/2) = |(1/2 + 1)/(1/2 + 2)| = (3/2)/(5/2) = 3/5
Since f(2) ≠ f(1/2), E is not the correct answer.
Answer: B
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