Polynomial and Rational Functions

Algebraic methods

Graphical methods

Analytical methods

Numerical methods

R(t) = E^{(t/5)} + 7

R(t) = E^{(t/5)} + 10

R(t) = E^{(t/5)} - 7

R(t) = E^{(t/5)} - 10

120 degrees

60 degrees

90 degrees

30 degrees

The graph touches but does not cross the y-axis at this point.

The graph has a maximum at this point.

The graph crosses the x-axis at this point.

The graph has an inflection point at this area.

H(x) = a(x + 5)^2 + 7

H(x) = -a(x + 5)^2 - 7

H(x) = a(x - 5)^2 - 7

H(x) = -a(x - 5)^2 + 7

An interval where $f$ passes both vertical and horizontal line tests ensuring it's both injective (one-to-one) and surjective (onto).

Zero coefficients for at least half of the terms in $f$ to reduce complexity for inversion purposes.

Symmetry about $y$-axis or origin so as to confine it within a range where inversion may occur effectively.

Exactly one

Two or four only

4

Unlimited

Zero only if both are even integers.

An integer only if both are multiples of each other.

An imaginary number only if both are positive integers.

A nonzero real number.

c = 2

c = -8

c = 0

c = -64

Inverses only exist for monomials and binomials due to their simpler structure.

Their leading coefficients prevent them from being inverted analytically.

They fail the horizontal line test because they are not one-to-one over their domains.

Polynomials never intersect themselves, so they always have functional inverses.