# Real-valued Functions and Algebra of Real-valued Functions

A function ƒ: A➞R which associates each element of set A with a unique real number ƒ(a) of set B is called a real-valued function.

If x is any real number, then the absolute value function defined by $ƒ(x)=|x|=\left\{\begin{array}{c} x, \text{ if } x≥0 \\ -x, \text{ if } x<0 \end{array}\right.$ is a real-valued function. Its graph is,

Consider two real-valued functions that have the same domain D. Suppose ƒ: D➞R and g: D➞R are two real-valued functions and k is a real number. Then, each of the following functions on the left side is defined by the formula on the right: $(ƒ+k):D➞R \text{ by } (ƒ+k)(x)=ƒ(x)+k$ $(|ƒ|):D➞R \text{ by } (|ƒ|)(x)=|ƒ(x)|$ $(ƒ^n):D➞R \text{ by } (ƒ^n)(x)=(ƒ(x))^n$ $(ƒ±g):D➞R \text{ by } (ƒ±g)(x)=ƒ(x)±g(x)$ $(kƒ):D➞R \text{ by } (kƒ)(x)=k(ƒ(x))$ $(ƒg):D➞R \text{ by } (ƒg)(x)=ƒ(x)g(x)$ $\left(\frac{ƒ}{g}\right):D➞R \text{ by } \left(\frac{ƒ}{g}\right)(x)=\frac{ƒ(x)}{g(x)} \text{ (}g(x)≠0)$ Examples:

1. Let ƒ: R➞R be defined by $ƒ(x)=x+3$ Then, $(ƒ+3)(x)=ƒ(x)+3$$=(x+3)+3$$=x+6$ $(ƒ^2)(x)=(ƒ(x))^2$ $=(x+3)^2$ $=x^2+6x+9$
2. Let ƒ: R➞R and g: R➞R be defined by $ƒ(x)=3x-1$ $g(x)=x^2$ Then, $(3ƒ-2g)(x)=3ƒ(x)-2g(x)$ $=3(3x-1)-2x^2$ $=-2x^2+9x-3$ $(ƒg)(x)=ƒ(x)g(x)$ $=(3x-1)x^2$ $=3x^3-x^2$

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