1.2 Combining Functions; Shifting and Scaling graphs

本文为《Thomas’ Calculus Early Transcendentals》阅读笔记
In this section we look at the main ways functions are combined or transformed to form new functions.

Sums, Differences, Products and Quotients

If $f$ and $g$ are functions, then for every $x$ that belongs to the domains of both $f$ and $g$ (that is, for $x\in D(f)\bigcap D(g)$ ), we define functions $f+g,f-g$ and $fg$ by the formulas

$$\begin{array}{rl}(f+g)(x)& =f(x)+g(x)\\ (f-g)(x)& =f(x)-g(x)\\ (fg)(x)& =f(x)g(x).\end{array}$$

At any point of $D(f)\bigcap D(g)$ at which $g(x)̸=0$, we can define the function $f/g$ by the formula

$$\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$$

EXAMPLE 1            The functions defined by the formulas $f(x)=\sqrt{x}$ and $g(x)=\sqrt{1-x}$

Composite Functions

DEFINITION           If $f$ and $g$ are functions, the composite function $f\circ g$ ($f$ composited with $g$) is defined by

$$(f\circ g)(x)=f(g(x)).$$

The domain of $f\circ g$ consists of the numbers $x$ in the domain for which $g(x)$ lies in the domain of $f$.

Shifting a graph of a function

Scaling and Reflecting a graph of a function