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# Domain

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Suppose the domain of f is (-1,1). Define the function l by l(x) = f((x - 1)/(x - 3)). What is the domain of l?

Apr 22, 2022

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For l(x) to make sense, it is required that $$\dfrac{x - 1}{x - 3} \in (-1, 1)$$.

Then, we solve the inequality $$-1 < \dfrac{x - 1}{x - 3} < 1$$ to find the domain of f.

I will do one side of the inequality as an example. Let's say we want to solve $$\dfrac{x - 1}{x - 3} < 1$$. We can't just multiply (x - 3) on both sides because it is not guaranteed that x - 3 is positive. Instead, we multiply (x - 3)^2 on both sides since it is always nonnegative. Also, note that x cannot be 3 because of the denominator.

$$\dfrac{x - 1}{x - 3} \cdot (x - 3)^2 \leq (x - 3)^2, \,x \neq 3\\ (x - 1)(x - 3) \leq (x - 3)^2, \, x \neq 3\\ x^2 - 4x + 3 \leq x^2 - 6x + 9,\, x \neq 3\\ 2x - 6 \leq 0, \, x \neq 3\\ 2x - 6 < 0\\ x < 3$$

Now, you can solve the other inequality $$-1 < \dfrac{x - 1}{x -3}$$ to get another inequality for x, combine them, and you get the domain of $$l(x)$$.

Apr 22, 2022