Elementary Functions at a Glance

Elementary functions are the building blocks of most formulas you will see in calculus and its applications.

Power Functions

• $a > 1$: grows faster; $a = 1$: line; $0 < a < 1$: concave, passes $(0,0)$.
• Negative $a$ yields reciprocals (e.g., $x^{-1} = \dfrac{1}{x}$, domain excludes $0$).

Exponential Functions

• Increasing if $a>1$, decreasing if $0<a<1$.
• Always positive, crosses $(0,1)$.

Logarithmic Functions

• Inverse of $a^x$; passes $(1,0)$.
• Increasing if $a>1$, decreasing if $0<a<1$.

Trigonometric Functions

• $\sin x$, $\cos x$ are bounded in $[-1,1]$ with period $2\pi$.
• $\tan x$, $\cot x$ have period $\pi$ and vertical asymptotes.
• $\sin$ is odd, $\cos$ is even; phase shifts create other shapes.

Inverse Trigonometric Functions

• $\arcsin x$: domain $[-1,1]$, range $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$
• $\arccos x$: domain $[-1,1]$, range $[0, \pi]$
• $\arctan x$: domain $\mathbb{R}$, range $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$

练习题

练习 1

State the domain and monotonicity of $f(x) = x^{1/2}$.

练习 2

Compare growth: which grows faster as $x \to +\infty$, $x^3$ or $2^x$?

练习 3

Give one even and one odd trigonometric function with their periods.

总结

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中英对照