Intervals & Real Functions
Closed Interval: Let a and b be two given real numbers such that a < b Then the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by [a, b].
i.e., [a, b] = {x ϵ R| a ≤ x ≤ b}.
Open Interval: Let a and b be two given real numbers such that a < b Then the set of all real numbers x such that a < x < b is called a closed interval and is denoted by (a, b).
i.e., (a, b) = {x ϵ R| a < x < b}.
Semi-Closed or Semi Open Interval: If a, b are two given real numbers such that a < b then the sets (a, b) = {x ϵ R| a < x ≤ b} and (a, b) = {x ϵ R|a ≤ x < b} are known as semi-closed or semi-open intervals and are also denoted by ]a, b[ and [a, b] respectively.
Real Function: If the domain and co-domain of a function are subsets of R (set of all real numbers). It is called a real valued function or in short, a real function.
Description of Areal Function: If f is a real valued function with finite domain, then f can be described by listing the values which it attains at different points of its domain. However, if the domain of a real function is an infinite set, then, f cannot be described by listing the values at points in its domain. In such cases real functions are generally described by some general formula or rule like f (x) = x² + 1 or f (x) = 2 sinx + 3 etc.
Example: If \(f\left( x \right)=x+\frac{1}{x}\), prove that \({{\left[ f\left( x \right) \right]}^{3}}=f\left( {{x}^{3}} \right)+3f\left( \frac{1}{x} \right)\).
Solution: We have,
\(f\left( x \right)=x+\frac{1}{x}\),
\(\Rightarrow f\left( {{x}^{3}} \right)={{x}^{3}}+\frac{1}{{{x}^{3}}}\).
Now,
\({{\left[ f\left( x \right) \right]}^{3}}={{\left( x+\frac{1}{x} \right)}^{3}}\),
\(=\left( {{x}^{3}}+\frac{1}{{{x}^{3}}} \right)+3\left( x+\frac{1}{x} \right)\),
\(=f\left( {{x}^{3}} \right)+3f\left( x \right)\),
\(=f\left( {{x}^{3}} \right)+3f\left( \frac{1}{x} \right)\,\left[ \because f\left( x \right)=f\left( \frac{1}{x} \right) \right]\).