Domain of a Function

Samia CHALLAL

28 Domain of a Function

$D_f$ is called the domain of $f$ : $D_f =\{ x\in \mathbb{R} : \, \, f(x) \in \mathbb{R}\}$

Example. If $P(x)$ and $Q(x)$ are polynomials, then

$\qquad\bullet\quad$ $f(x) = P(x) \quad \Longrightarrow \quad D_f = \mathbb{R}$

$\qquad\bullet\quad$ $f(x) = \displaystyle{\frac{P(x)}{Q(x)} } \quad \Longrightarrow \quad D_f =\{ x\in \mathbb{R} : \, \, Q(x) \neq 0 \}$

$\qquad\bullet\quad$ $f(x) = \sqrt{P(x) } \quad \Longrightarrow \quad D_f = \{ x\in \mathbb{R} : \, \, P(x) \geqslant 0 \}$

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Exercise 1

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We have $f(x) = \displaystyle{\frac{ 1 }{ 3 -\sqrt{x+1}}}$

$\displaystyle{D_f= \{ x \in \mathbb{R} : \qquad x+1\geqslant 0\quad\hbox{ and } \quad 3 -\sqrt{x+1}\neq 0\}= [-1, 8)\cup (8 , + \infty) }$

since we have

$3 -\sqrt{x+1}= 0 \quad \Longleftrightarrow \quad 9= x+ 1 \quad \hbox{ with } \quad x+1\geqslant 0$

$\quad \Longleftrightarrow \quad x=8 \quad \hbox{ with } \quad x\geqslant -1$

$\quad\Longleftrightarrow \quad x=8 .$

Exercise 2

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We have

$D_f= \{ x \in \mathbb{R} : \quad 2-\cos x\neq 0 \}= \mathbb{R}$

since

$-1 \leqslant \cos x \leqslant 1 \quad \Longleftrightarrow \quad -1 \leqslant -\cos x \leqslant 1$

$\quad \Longleftrightarrow \quad 1 \leqslant 2 -\cos x \leqslant 3$

$\quad \Longrightarrow \quad 2-\cos x\neq 0 \quad \forall x\in \mathbb{R} .$

Exercise 3

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We have $f(x) = \displaystyle{\sqrt{\frac{x- 3x^2}{x+1}}}$

$\displaystyle{D_f= \{ x \in \mathbb{R} : \quad \frac{x- 3x^2}{x+1}\geqslant 0\quad\hbox{ and } \quad x+1\neq 0\}= (- \infty, -1)\cup [0 , 1/3] }$

since we have

$\begin{array}{|c|ccccccccc|}\hline x & -\infty & \hskip 1cm & -1 & \hskip 1cm & 0 & \hskip 1cm & 1/3 & \hskip 1cm & +\infty \\ \hline x(1-3x)& & - & & - & & + & & - & \\ \hline x+1 & & - & & + & & + & & + & \\ \hline \displaystyle{\frac{x- 3x^2}{x+1}} & & + & & - & & + & & - & \\ \hline \end{array}$

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