初等函数

羊男原创2025/6/10大约 3 分钟

幂函数

Power Function

f (x) = x^{M}, M \in R

f (x) = x

f (x) = x^{2}

f (x) = x^{3}

f (x) = x^{- 1} = \frac{1}{x}

f (x) = x^{- 2} = \frac{1}{x^{2}}

f (x) = x^{\frac{1}{2}} = \sqrt{x}

f (x) = x^{\frac{1}{3}} = \sqrt[3]{x}

Exponential Function

f (x) = a^{x}, (a > 0, a \neq 1)

f (x) = a^{x}, 0 < a < 1

a = 0.5

f (x) = a^{x}, a > 1

a = 2

f (x) = e^{x}

$e^{1} = e$

$a^{n} \times a^{m} = a^{n + m}$

$a^{n} \div a^{m} = a^{n - m}$

$(a^{n})^{m} = a^{n m}$

$(a b)^{n} = a^{n} b^{n}$

Logarithmic Function

f (x) = \log_{a} x, (x > 0, a > 0, a \neq 1)

f (x) = \log_{a} x, 0 < a < 1

a = 0.5

f (x) = \log_{a} x, a > 1

a = 2

f (x) = \ln x = l o g_{e} x

$\ln e = 1$
$\ln \frac{1}{e} = \ln e^{- 1} = - 1$

$\log_{a} (M N) = \log_{a} M + \log_{a} N$

$\log_{a} \frac{M}{N} = \log_{a} M - \log_{a} N$

$l o g_{a} M^{n} = n l o g_{a} M$

$\log_{a^{n}} b^{m} = \frac{m}{n} l o g_{a} b$

$\log_{a} b = \frac{\log_{c} b}{\log_{c} a}$ （换底公式）

Trigonometric Functions

f (x) = \sin x

最小正周期： $2 π, 0 ∽ 2 π$

f (x) = \cos x

最小正周期： $2 π, 0 ∽ 2 π$

f (x) = \tan x

最小正周期： $π, - \frac{π}{2} ∽ \frac{π}{2}$

$\tan x = \frac{\sin x}{\cos x}$

f (x) = \cot x

最小正周期： $π, 0 ∽ π$
$\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$

f (x) = \sec x

$\sec x = \frac{1}{\cos x}$

f (x) = \csc x

$\csc x = \frac{1}{\sin x}$

	$0 °$	$30 °$	$45 °$	$60 °$	$90 °$
		$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
$\sin$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan$	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$
$\cot$		$\sqrt{3}$	$1$	$\frac{\sqrt{3}}{3}$	$1$

用到再补充吧

Inverse Trigonometric Function

因为，函数必须是单射的，所以反三角函数函数需要限定区间

f (x) = \arcsin x

$f (x) = \sin x, x \in [- \frac{π}{2}, \frac{π}{2}]$ 的反函数为

$f (x) = \arcsin x, x \in [- 1, 1]$

f (x) = \arccos x

$f (x) = \cos x, x \in [0, π]$ 的反函数为

$f (x) = \arccos x, x \in [- 1, 1]$

f (x) = \arctan x

$f (x) = \tan x, x \in (- \frac{π}{2}, \frac{π}{2})$ 的反函数为
$f (x) = \arctan x, x \in (- \infty, \infty), f (x) \in (- \frac{π}{2}, \frac{π}{2})$

f (x) = a r c c o t x

$f (x) = \cot x, x \in (0, π)$ 的反函数为
$f (x) = a r c c o t x, x \in (- \infty, \infty), f (x) \in (0, π)$