List of Notes - Category: Common Functions
Linear function is given as `y=f(x)=mx+b`.
`m` is called slope and `b` is called y-intercept. Graph of the linear function is line. Since there are two parameters in the linear function (m and b) it is enough two points to uniquely identify line.
Polynomial is a function of the form `y=f(x)=a_n x^n+a_(n-1) x^(n-1)+...+a_2 x^2+a_1 x+a_0` , where n is nonnegative integer and `a_0,\ a_1,\ ...,\ a_(n-1),\ a_n` are constants which are called coefficients of polynomial.
Function of the form `f(x)=x^n` where `n` is constant is called power function.
Depending on value of `n` graph of power function has different forms and properties.
Case 1: `n` is positive integer.
Function of the form `f(x)=(Q(x))/(P(x))=(a_0x^n+a_1x^(n-1)+...+a_(n-1)x+a_n)/(b_0x^m+b_1x^(m-1)+...+b_(m-1)x+b_m)`, where `Q(x)` and `P(x)` are polynomials is called rational function.
Domain of this function consists of all `x` such that `Q(x)!=0`.
A function is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) on polynomials.
Any rational function is automatically an algebraic function.
Function of the form `f(x)=a^x` where `a>0` is called exponential function.
Do not confuse it with power function `f(x)=x^a` in which the variable is the base.
Domain of the exponential function is `(-oo,oo)`, range is `(0,oo)`, provided `a!=1`.
Since exponential function `y=a^x` is monotonic (i.e. either increasing or decreasing) then it passes horizontal line test. Thus, it has inverse function. To find this function we use algorithm for finding inverse.
Consider unit circle centered at origin and point `P_0(1,0)`. If we begin to rotate point `P_0` around origin on angle `t` then we will obtain point `P_t`.
x-coordinate of this point is called cosine of number `t` and denoted by `cos(t)`, y-coordinate of this point is called sine of number `t` and denoted by `sin(t)`.
Clearly all trigonometric functions due to periodicity fail horizontal line test. Therefore, they don't have inverse. But if we consider interval where piece of function passes horizontal line test and takes all values from its range, then we can find inverse function.
Hyperbolic cosine is `color(red)(y=cosh(x)=(e^x+e^(-x))/2)`.
Hyperbolic sine is `color(blue)(y=sinh(x)=(e^x-e^(-x))/2)`.
Hyperbolic tangent is `color(green)(y=tanh(x)=(sinh(x))/(cosh(x))=(e^x-e^(-x))/(e^x+e^(-x)))`.
Hyperbolic cosine is `y=cosh(x)=(e^x+e^(-x))/2`.
This function is not one-to-one, so there is no unique inverse for this function. However, if we take function on interval `(0,oo)` then it will have unique inverse.