Introduction to Hilbert space by Berberian S.K.

By Berberian S.K.

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Y = − a f a f ⇒ a x − 1f a x − 3f ≥ 0 ⇒ x−3 − x−3 ≥0 2 ⇒ x −4≤0 1 10. e; y = remember the following facts: af af f x ) one must g x 29 Function values of x for which af af af af f x ) consists of those g x 1. The domain of y ( = af af f x ≥0 g x af af af f x ≥ 0 ⇔ f x ≥ 0 , g x > 0 , or f x ≥ 0 , 2. g x g (x) < 0. 3. FG x − α IJ ≤ 0 ⇔ α ≤ x < β H x − βK or β < x ≤ α ac- cording as α < β or β < α . 4. FG x − α IJ ≥ 0 ⇔ x ≥ α or x < β if β < α and ⇔ H x −βK LM N Hence, domain = R − −3 , LM 2 , + ∞I N3 K I a K f 2 = −∞ , − 3 ∪ 3 or, alternatively: 2 x− 3x − 2 3 ≥ 0 ⇔ x < − 3 or ≥0⇔ x+3 2x + 6 f LMN IK 2 Hence, domain = a−∞ , − 3f ∪ L , + ∞I MN 3 K x≥ a 2 2 ⇔ x ∈ −∞ , − 3 ∪ ,+∞ 3 3 x −1 x +1 x ≤ α or x > β if α < β .

2. When a function defined by a single formula y = f (x) does not become imaginary or undefined for any value of independent variable x, the domain of the function y = f (x) is the set of all real numbers denoted by R. To obtain its range, one should consider the domain −∞ < x < ∞ using the axioms of inequality in −∞ < x < ∞ . 3. , 2 D = b − 4 ac ≥ 0 for real x. 4. In case the domain of a function y = f (x) is a finite set D = {a1, a2, a3, …, an}, then its range is obtained by forming the set whose members are the values of [f (x)] x = a1, a2, a3, …, an.

999 ∈ [–1, 0) = −1 ≤ x < 0 ∴ f (–1 + h) = (1 + x)x =–1 + h = 1 + h – 1 = h Domain of a Function Sometimes a function of an independent variable x is described by a formula or an equation or an expression in x and the domain of a function is not explicitly stated. In such circumstances, the domain of a function is understood to be the largest possible set of real numbers such that for each real number (of the largest possible set), the rule (or, the function) gives a real number or for each of which the formula is meaningful or defined.

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