当前位置：网站首页>Power function Exponential function Logarithmic function

Power function Exponential function Logarithmic function

2022-08-10 06:50:00 【Ye Xian】

幂函数
$\f Use macros to define as f(#1) y = x^u$ (u为实数)

幂函数 $\f Use macros to define as f(#1) y = x^u$ The domain and value range of u的取值,当x>0时, $\f Use macros to define as f(#1) y = x^u$ 都有定义.
Common Power Functions：y=x, $\f Use macros to define as f(#1) y = x^2$ , $\f Use macros to define as f(#1) y = x^3$ $\f Use macros to define as f(#1) y = \sqrt[3]{x} , y=\sqrt{x},y=\frac{1}{x}.$

指数函数
$\f Use macros to define as f(#1) y=a^x$ (a>0,a $\not =$ 1)

定义域：（- $\infty$ ,+ $\infty$ ）,值域：（0,+ $\infty$ ）.
单调性：当a>1时, $y=a^x$ 单调增;当0<a<1时, $y=a^x$ 单调减.
Common exponential function： $y=e^x$ ,单调增, $\lim \atop x\rightarrow-\infty$ $e^x =0$ , $\lim \atop x\rightarrow +\infty$ $e^x =+\infty$ .
部分解释：a $\not =$ 1就不用说了,a=1就全是1了 ;a>0,如果a<0,比如a= -5,那么便是 $y=(-5)^x$ , x可以为 $\frac{1}{2}$ ,即 $\sqrt{-5}$ ,Obviously, the square root cannot be zero,Zero is meaningless; $0^0=1,0^n=0(n>0),0^-\raisebox{0.25em}{n} =\frac{1}{0^n}(n>0)(And the denominator cannot be0的)$

对数函数

$\raisebox{0.25em}{y} \raisebox{0.25em}{=} \raisebox{0.25em}{log} a \raisebox{0.25em}{x}$ (a>0,a $\not =$ 1)

定义域：（0,+ $\infty$ ）,值域：（- $\infty$ ,+ $\infty$ ）.
单调性：当a>1时, $\raisebox{0.25em}{y} \raisebox{0.25em}{=} \raisebox{0.25em}{log} a \raisebox{0.25em}{x}$ 单调增;当0<a<1时, $\raisebox{0.25em}{y} \raisebox{0.25em}{=} \raisebox{0.25em}{log} a \raisebox{0.25em}{x}$ 单调减.
Common logarithmic functions： $y = l n x$ ,单调增, $\lim \atop x\rightarrow-\infty$ $lnx=-\infty$ , $\lim \atop x\rightarrow +\infty$ $=+\infty$ .
部分解释：如果a不大于0,可能出现一个x对应多个y值,This doesn't fit the definition of a function,如a=-1, $-1\raisebox{0.25em}{y}=x$ ,如x=1,There are countlessy与其对应

版权声明
本文为[Ye Xian]所创，转载请带上原文链接，感谢
https://yzsam.com/2022/222/202208100638336010.html