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Elementary Functions

Constant Functions​

f(x)=c,c∈R.\begin{equation*} f(x) = c, \quad c \in \mathbb{R}. \end{equation*}

  • Domain: R\mathbb{R}.

  • Range: {c}\{c \}.

Identity Function​

f(x)=x.\begin{equation*} f(x) = x. \end{equation*}

  • Domain: R\mathbb{R}.

  • Range: R\mathbb{R}.

Exponential Functions​

f(x)=ax,a>0,  a≠1.\begin{equation*} f(x) = a^x, \quad a>0,\; a\neq1. \end{equation*}
  • Domain: R\mathbb{R}.
  • Range: (0,+∞)(0, +\infty).
  • Special case: f(x)=exf(x) = e^x is the natural exponential function.

Logarithmic Functions​

f(x)=log⁡ax,a>0,  a≠1.\begin{equation*} f(x) = \log_a x, \quad a>0,\; a\neq1. \end{equation*}
  • Domain: (0,+∞)(0, +\infty).
  • Range: R\mathbb{R}.
  • Special case: f(x)=ln⁡x=log⁡exf(x) = \ln x = \log_e x.
  • Inverse: (log⁡ax)−1=ax.(\log_a x)^{-1} = a^x.

Power Functions​

f(x)=xa=ealn⁡x,a∈R.\begin{equation*} f(x) = x^a = e ^ {a \ln x}, \quad a \in \mathbb{R}. \end{equation*}
  • Domain: (0,+∞)(0, +\infty).
  • Range: (0,+∞)(0, +\infty).
  • Typical examples: x2,  x3,  x,  1xx^2,\; x^3,\; \sqrt{x},\; \dfrac{1}{x}.

Trigonometric Functions​

sin⁡x,  cos⁡x,  tan⁡x=sin⁡xcos⁡x,  cot⁡x=cos⁡xsin⁡x,  sec⁡x=1cos⁡x,  csc⁡x=1sin⁡x.\begin{equation*} \sin x,\; \cos x,\; \tan x = \frac{\sin x}{\cos x},\; \cot x = \frac{\cos x}{\sin x},\; \sec x = \frac{1}{\cos x},\; \csc x = \frac{1}{\sin x}. \end{equation*}
  • Domain and Range:
FunctionDomainRange
sin⁡x\sin xR\mathbb{R}[−1,1][-1,1]
cos⁡x\cos xR\mathbb{R}[−1,1][-1,1]
tan⁡x\tan xx≠π/2+kπx\neq \pi/2 + k\piR\mathbb{R}

Theorems:​

  1. sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\\.

  2. cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\\.

  3. tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B.\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}.

  4. sin⁡(2A)=2sin⁡Acos⁡A\sin(2A)=2\sin A\cos A.

  5. cos⁡(2A)=cos⁡2A−sin⁡2A=2cos⁡2A−1=1−2sin⁡2A\cos(2A)=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A.

  6. tan⁡(2A)=2tan⁡A1−tan⁡2A\tan(2A)=\frac{2\tan A}{1-\tan^2A}.

  7. sin⁡A+sin⁡B=2sin⁡A+B2cos⁡A−B2\sin A+\sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}.

  8. sin⁡A−sin⁡B=2cos⁡A+B2sin⁡A−B2\sin A-\sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}.

  9. cos⁡A+cos⁡B=2cos⁡A+B2cos⁡A−B2\cos A+\cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}.

  10. cos⁡A−cos⁡B=−2sin⁡A+B2sin⁡A−B2\cos A-\cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}.

  11. sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A\sin B = \tfrac{1}{2}\big[\cos(A-B)-\cos(A+B)\big].

  12. cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A\cos B = \tfrac{1}{2}\big[\cos(A-B)+\cos(A+B)\big].

  13. sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A\cos B = \tfrac{1}{2}\big[\sin(A+B)+\sin(A-B)\big].

  14. For angles θ\theta in the interval (0,π2):(0,\tfrac{\pi}{2}): sin⁡θ<θ<tan⁡θ\sin\theta < \theta < \tan\theta.

Inverse Trigonometric Functions​

arcsin⁡x,arccos⁡x,arctan⁡x.\begin{equation*} \arcsin x,\quad \arccos x,\quad \arctan x. \end{equation*}
  • Domains and ranges:
FunctionDomainRange
arcsin⁡x\arcsin x[−1,1][-1,1][−π/2, π/2][-\pi/2,\ \pi/2]
arccos⁡x\arccos x[−1,1][-1,1][0, π][0,\ \pi]
arctan⁡x\arctan xR\mathbb{R}(−π/2, π/2)(-\pi/2,\ \pi/2)

Operations and Compositions​

If f,gf,g are elementary functions, then

f+g,f−g,f⋅g,fg  (g≠0),f∘g\begin{equation*} f+g,\quad f-g,\quad f\cdot g,\quad \frac{f}{g}\;(g\neq0),\quad f\circ g \end{equation*}

are all elementary functions (in their domains of definition).

Example:

f(x)=sin⁡x,g(x)=ex⇒(f∘g)(x)=sin⁡(ex)\begin{equation*} f(x)=\sin x,\quad g(x)=e^x \Rightarrow (f\circ g)(x)=\sin(e^x) \end{equation*}