Honors Calculus I - 3 Function

Function

For $f:X\to Y$ is a mapping

• domain : $D_f=X$
• codomain : $Y$
• range : $R_f$ (The definite numbers)
• $y$ is the image of $x$
• $x$ is the inverse image of $y$

If :

• $R_f=Y$, that $f$ is a surjective
• $f(x_1)=f(x_2)$ implies $x_1=x_2$, that $f$ is a injective
• bijective : a mapping $f$ is both surjective and injective.

Inverse mapping

Where $f^{-1}:R_f\to X$. A mapping is inversible iff it is bijective.

Cancellation Equations :

• $f^{-1}(f(x))=x$
• $f(f^{-1}(x))=x$

The inverse function $f^{-1}(x)$ is the reflection of that of the function $f(x)$ with respect to the diagonal line $y=x$

• Composition function : $h(x)=g(f(x))$, or $h=g\circ f$

Intervals

• Neighborhood : $N_r(p)=\{q\in \R:|p-q|<r\}$, where $r$ is called the radius
• Interior point of E: $N_r(p)\subset E$
• monotonically increasing / decreasing
• odd / even function
• periodic function

Basic Elementary Function

Power function

For fixed $n\in\N$, $f(x)=x^n$

Radical Function

For fixed $n\in \N_{+}$, $f(x)=\sqrt[n]x$

• When $n$ is even, the domain and range are both $[0,+\infty)$
• While $n$ is odd, the domain and range are $(-\infty,+\infty)$

Exponential Function

For $a>0$, $f(x)=a^x$

• When $0<a<1$, $f(x)$ is strictly decreasing in its domain; when $a>1$, $f(x)$ is strictly increasing in its domain
• laws of exponents

Logarithmic function

$f(x)=\log_ax$

• Laws of logarithms
• natural logarithm : $y=\ln x$

Trigonometric functions

sine function, cosine function and tangent funtion

Two key equalities hold (which almost all basic trigonometric identities can be derived)

• Pythagorean identity : $\sin^2x+\cos^2x=1$
• Euler’s formula : $e^{ix}=\cos x+i\sin x$

• cotangent: $\cot \theta = \dfrac 1 {\tan \theta}$
• secant : $\sec \theta =\dfrac 1{\cos \theta}$
• cosecant : $\csc\theta =\dfrac 1 {\sin \theta}$

• $\theta = \arcsin y\; (\sin^{-1}y)$ : where $y$ is in $[-1,1]$ and $\theta$ in $[-{\pi\over 2},{\pi\over 2}]$
• $\theta = \arccos y\; (\cos^{-1}y)$ : where $y$ is in $[-1,1]$ and $\theta$ in $[0,\pi]$
• $\theta = \arctan y\; (\tan^{-1}y)$ : where $y$ is in $(-\infty,+\infty)$ and $\theta$ in $(-{\pi\over 2},{\pi\over 2})$