{"id":4293,"date":"2013-12-12T06:05:35","date_gmt":"2013-12-11T21:05:35","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4293"},"modified":"2014-08-01T19:22:16","modified_gmt":"2014-08-01T10:22:16","slug":"special-types-of-functions","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4293","title":{"rendered":"Special types of functions"},"content":{"rendered":"
\n

Polynomials<\/h3>\n

Polynomial is formula as below;<\/p>\n

\\displaystyle f(x) = a_0x^n + a_1x^{n-1} + a_2x^{n-2} + \\cdots + a_n<\/p>\n

If a_0 \\neq 0, n is called as degree of polynomials. <\/p>\n

\\displaystyle (a + x)^n = a^n + \\left(\\frac{n}{1}\\right)a^{n -1}x + \\left(\\frac{n}{2}\\right)a^{n -2}x^2 + \\cdots + x^n<\/p>\n

where the binomial coefficients are given by<\/p>\n

\\displaystyle \\left(\\frac{n}{k}\\right) = \\frac{n!}{k!(n - k)!}
\nand where factorial n, i.e. n! = n(n -1)(n-2)…1 while 0! = 1 by definition.<\/p>\n

Exponential function<\/h3>\n

\\displaystyle f(x) = a^x<\/p>\n

An important special case occurs where a = e = 2.718…<\/p>\n

Exponential law<\/h4>\n
    \n
  1. \\displaystyle a^{m + n} = a^m \\cdot a^n<\/li>\n
  2. \\displaystyle a^{m - n} = \\frac{a^m}{a^n},\\ a \\neq 0<\/li>\n
  3. \\displaystyle (a^m)^n = a^{mn}<\/li>\n<\/ol>\n

    Logarithmic function<\/h3>\n

    \\displaystyle f(x) = \\log_a x<\/p>\n

    These functions are inverse of the exponential functions, i.e. if ax<\/sup> = y then x = loga<\/sub>y where a is called the base of the logarithm. If a = e, which is often called the natural base of logarithm, it’s described loge<\/sub> by ln x, called the natural logarithm of x.<\/p>\n

    Logarithmic law<\/h4>\n
      \n
    1. \\displaystyle \\ln(mn) = \\ln(m) + \\ln(n)<\/li>\n
    2. \\displaystyle \\ln\\frac{m}{n} = \\ln(m) - \\ln(n)<\/li>\n
    3. \\displaystyle \\ln{m^p} = p\\ln{m}<\/li>\n<\/ol>\n<\/blockquote>\n