Solution please?
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They’re basically asking for the range of values x + \frac{1}{x} or \frac{x^2 + 1}{x} can have.
As you can see, for numbers greater than 1, the numerator is always greater than the denominator, and the result gets bigger and bigger as you the x increases, so all values above 2 are possible (when x starts from 1 and increases).
For numbers between 0 and 1, as it goes to 0, the result will again increase towards infinity.
Now for negative numbers, only the sign changes but the behaviour of the result is the same.
Let’s recap, coming from -\infty towards 0, the value increases from -\infty to -2 ( -2 is achieved at x = -1) and again starts decreasing towards -\infty as x gets closer to zero. Similar results for +ve values of x. So the range basically becomes (-\infty,-2] or [2,+\infty).
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Another approach would be to assume the expression can have a solution k. So, \frac{x^2 + 1}{x} = k. Simplify to get a quadratic equation. x^2 -kx + 1 = 0
Now the formula for the roots of a quadratic equation is
So to find what values k can have, just make sure that whatever is under the square root is not negative (wiz k^2 - 4). So for k^2 - 4 to not be negative, k must be ≥ 2 or ≤-2. Same solution, different concept.
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