types of functions

parent function

a function in its most basic form that shows the relationship between the independent and dependent variables in their pre-transformed state

even functions

Any function that has symmetry with respect to the y-axis; algebraically: f(−x) = f(x), aka all x values become negative

odd functions

Any function that has rotational symmetry of 180° about the origin; Algebraically: 𝑓(−𝑥) = −𝑓(𝑥), aka both x and y values become negative

𝑓(𝑥) = 𝑏, 𝑏 ∈ ℛ

constant function: horizontal line, even, not 1 to 1

𝑓(𝑥) = 𝑥

linear function: odd, 1 to 1, straight positive line

𝑓(𝑥) = 𝑥²

square function: quadratic, makes parabola, even, not 1 to 1

𝑓(𝑥) = √𝑥

square root/radical function: has a starting point, curves slightly, extends infinitely; neither even or odd, 1 to 1

𝑓(𝑥) = 𝑥³

cubic function: horizontal “S” shape, odd, 1 to 1, no decreasing

𝑓(𝑥) = ³√𝑥

cube root function: “S” shape, odd, 1 to 1, no decreasing

𝑓(𝑥) = 1/x

reciprocal/rational function: two corner asymptotes at x=0 and y=0, two domains/ranges, odd, 1 to 1

𝑓(𝑥) = 1/x²

squared reciprocal function: 2 asymptotes abt to join together, even, 1 to 1

𝑓(𝑥) = |𝑥|

absolute value function: “V” shape, even, not 1 to 1

𝑓(𝑥) = 𝑏^x, 𝑤ℎ𝑒𝑟𝑒 𝑏 > 0, 𝑏 ≠ 1, 𝑜𝑟 𝑓(𝑥) = 𝑒^x

exponential function: 1 asymptote at y=0 going up hill, neither even or odd, 1 to 1

𝑓(𝑥) = log_b𝑥, 𝑤ℎ𝑒𝑟𝑒 𝑏 > 0, 𝑏 ≠ 1, 𝑜𝑟 𝑓(𝑥) = ln𝑥

logarithmic function: 1 asymptote at x=0 + looks similar to square root function but extends down x axis and to the right corner, neither even or odd, 1 to 1

vertical asymptote

x=0 → graph gets infinitely closer to the y axis

horizontal asymptote

y=0 → graph gets infinitely closer to the x axis

𝑓(𝑥) = sin𝑥

sine function:

𝑓(𝑥) = cos𝑥

cosine function:

𝑓(𝑥) = [𝑥] 𝑜𝑟 𝑓(𝑥) = 𝑖𝑛𝑡(𝑥)

greatest integer function: