Math

• Function


  A relation in which each element of the domain corresponds to exactly one element of the range; a relation where each element in the domain is related to only one value in the range by some rule; a set of ordered pairs (x,y) such that no two ordered pairs have the same x-value but different y-values
• Function


  • Using functional notation, we can write f(x) = y, read as "f of x is equal to y"
  • If (1, 2) is an ordered pair associated with the function f, then we say that f(2) = 1
• Functions


  • (1, 2), (2, 3), (3, 4), (4, 5)
  • (1, 3), (0, 3), (2, 1), (4, 2)
  • (1, 6), (2, 5), (1, 9), (4, 3)
• Domain


  The set of the first coordinates or the x (the set D is the domain of f)
• Domain


  • The denominator (bottom) of a fraction cannot be zero
  • The number under a square root sign must be positive
• Domain


  • y = √x+4, the domain is x ≥ -4
• Range


  The set of the second coordinates or the y (the set R is the range of f)
• Range


  • The range of a function is the spread of possible y-values (minimum y-value to maximum y-value)
  • Substitute different x-values into the expression for y to see what is happening
• Range


  • y = √x+4, the range is y ≥ 0
• Relation


  A rule that relates values from a set of values (called the domain) to a second set of values (called the range); a set of ordered pairs (x,y)
• Relation


  • Relation in table, graph, and mapping diagram
• Piecewise Function


  Known as a compound function; defined by multiple sub-functions where each sub-function applies to a certain interval of the main function's domain
• Piecewise Function


  • h(x) = 2 for x ≤ 1, h(x) = x for x > 1
• Floor Function


  A very special piecewise function with an infinite number of pieces
• Evaluating a Function


  Finding the value of the function for a given value of the variable
• Evaluating a Function


  • f(x) = x^2 - 5x + 3, f(4) = 16 - 20 + 3 = -1
• Operations on Functions


  Sum, Difference, Product, Quotient
• Operations on Functions


  • (g - f)(x)
  • (g/h)(x) where h(x) ≠ 0
• Composition of Functions


  Another method of combining functions
• h(4)


  4
• x is > 1
• h(x)


  x
• Operations on a Function

  SUM: (g + h)(x)= g(x)+h(x)<|>DIFFERENCE: (g-h)(x)= g(x)-h(x)<|>PRODUCT: (gh)(x)= g(x) . h(x)<|>QUOTIENT: (g/h)(x)= g(x)/h(x); h(x) # 0
• Composition of Functions


  Another method of contributing a function from two given functions; consists of using the range element of one function as the domain element of another function
• Rational Equations


  Equations that contain rational expressions; can be solved using the techniques for performing operations with rational expressions and for solving algebraic equations
• Rational Function


  A function of the form f(x)= P(x)/Q(x) where P and Q are polynomials. The domain consists of all real numbers x except those for which the denominator is zero.
• R(x) = (x^2 + 4x - 1) / (3x^2 - 9x + 2)

  • R(x) = 1 / ((x - 1)(x^2 + 3))
• Rational Inequality


  An inequality which contains a rational expression. The trick is to always work with zero on one side of the inequality.
• Vertical Asymptote


  The line x=a of the function y=f(x) if y approaches (pos.&neg.)infinity as x approaches a from the right or left
• Horizontal Asymptote


  The line y=b of the function y=f(x) if y approaches b as x approaches (pos.&neg. infinity)
• Rules for finding Horizontal Asymptotes

  • If both polynomials are the same degree, divide the coefficients of the highest degree terms.
  • If the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote.
  • If the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote. There is a slant asymptote instead.
• Theorem on Vertical Asymptote

  If the real number a is a zero of the denominator Q(x), then the graph of f(x)= P(x)/Q(x), where P(x) and Q(x) have no common factors, has the vertical asymptote x=a
• Oblique Asymptote


  Also known as diagonal or slant asymptote; the line y=mx+b is an oblique asymptote for the graph of f(x) if f(x) gets close to mx+b or x gets really large or really small