ch1

  • lnx → x, raise the whole thing to the power of e

  • if the function involves square roots, remember that the square root of a negative number does not exist so take that into consideration when finding the domain

    • in the case of f(g(x)), where g(x) is a square root, to find the domain of the final function the domain of g(x) must be taken into consideration and the final domain cannot include functions that are not allowed in g(x)

    • this does not apply to other fractional functions, in that case just find the domain normally and also no need to simplify (kcf)

  • when finding the domain for ln functions as fractions, always remember that if both the numerator and denominator are negative, the final answer is positive so it might be allowed

    • draw a table like the pos/neg ones and see

    • the domain might be a union of two domains

  • in piecewise functions, the domain is lowk always given so it’s just the union of the domains of the functions in the piecewise function

  • the domain of multiplication of two functions is what’s shared between the domains of both functions

  • remember that absolute value functions have two outcomes, positive and negative

  • when given the domain of f(x) being [this, that], to find the domain of f(a function) put:

    • this < a function < that

    • then simplify until the function becomes just x and that gives you the domain

  • when finding the range of a function, use the whole function same way u do for x in the domain

    • then multiply and add and whatever

  • e^x > 0 always

    • its domain is R

  • TRIG:

    • sin,

      • domain = [-pi/2 , pi/2]

      • range = [-1,1]

    • cos,

      • domain = [0,pi]

      • range = [-1,1]

  • domain of the inverse = range

  • shifting:

    • LEFT = f(x+a)

    • RIGHT = f(x-a)

    • UP = f(x) + a

    • DOWN = f(x) - a

  • reflection:

    • ABOUT THE Y AXIS = changing x → (-x,y)

    • ABOUT THE X AXIS = changing y → (x,-y)

  • shrinking:

    • VERTICAL = kf(x)

    • HORIZONTAL = f(xk)

  • don’t shift and reflect at the same time, shift then reflect

  • EVEN AND ODD FUNCTIONS

    • even: f(-x) = f(x)

      • even functions are symmetric about the y axis

    • odd: f(-x) = -f(x)

      • odd functions are symmetric about the origin

  • sq(x²) = |x|

  • (sq(x))² = x

  • ax²+bx+c

    • to find x after altering the function, use x=-b/2a

    • find (x,y) of both the new and old functions and to find out what happened (shifting, reflecting, etc) do NEW-OLD to get the differences in the (x,y) coordinates

  • if two logarithmic functions have the same base (log 5s or lns for example), subtraction means division and addition means multiplication

    • same base and other number means u can combine their exponents

  • ify = logb(x)

    • then b^y = x

  • quadratic formula: ( -b +- sq( b²-4ac ) ) / 2a

  • loga(b) = lnb/lna

  • to cancel out e^-x and e^x, multiply both sides of the equation by e^x