Algebra and functions

• a^s x a^t = a^(s + t) and a^s / a^t = a^(s - t).
• (a^s)^t = a^(s x t).
• (ab)^s = (a^s)(b^s).
• a^0 = 1 and a^1 = a.
• a^-s = 1/ a^s.
• a^(1/s) = ^s root(a) and a^(s/t) = ^t root(a^s).
• The difference of two squares: x^2 - y^2 = (x + y)(x - y).
• root(ab) = root(a) x root(b) and root(a/b) = root(a) / root(b).
• The quadratic formula is x= (-b +/- root(b^2 - 4ac))/2a , where y = ax^2 + bx + c = 0.
• Completing the square is putting ax^2 + bx + c in the form a (x/a + b/2a)^2 - (b/2a)^2 + c.
• The discriminant of a function is b^2 - 4ac and if it is more than zero, the function has two real roots, equal to zero, it has one repeated real root and if it is less than zero, it has no real roots.
• To solve a linear inequality, find the set of all real numbers that make the inequality valid.
• When plotting an equality on the number line, the boundaries are marked by a dot with a filled in dot indicating a more/less than OR equal to inequality and an unfilled dot indicating just a more/less than inequality.
• When an inequality is more than zero, the values above the x axis are valid, and when an inequality is less than zero, the values below the x axis are valid.
• The translation of y = f(x) + a can be represented by the vector 0 i + a j and the translation of vector y = f(x+a) can be represented by the vector -a i + 0 j.
• y = af(x) stretches the graph in the vertical direction by a multiple of a and y = f(ax) stretches the graph in the horizontal direction by a multiple of 1/a.
• y = -f(x) is the reflection of y = (fx) in the x axis and y = f(-x) is the reflection of y = f(x) in the y axis.
• If f(x) is a polynomial then: If f(a) = 0, (x - a) is a factor of f(x) and a is a root.
• Any even number, n, can be written in the form n = 2k for some integer k, and any odd number, n, can be written in the form n = 2k+1 for some integer k.
• For an improper fraction, you must perform algebraic long division before it can be split up into partial fractions.
• Whenever there is a repeated linear factor in the denominator that needs to be split into partial fractions an extra fraction with the denominator as a single version of that factor must be used as well as the fraction with the denominator as the repeated factor.
• A one to one function is one where each input has a unique output, a many to one function has some inputs mapped to the same output, and a one to many function has inputs that are mapped to many outputs.
• Functions can only be one to one or many to one, not one to many mappings.
• The domain is the set of all possible inputs (x values) and the range is the set of all possible outputs (y values).
• The modulus of any number is its non negative value, and is sometimes called the absolute value.
• An inverse function is the original function reflected in the line y = x, and the domain of the original equals the range of the inverse and vise versa.
• To get an inverse function, interchange the x and y variables of the original function and make y the subject again.