Functions and Variations - Pre Calculus PDF

Functions and Variations -Pre Calculus- Raye Rhouieze B. Miranda Functions  Relations A relation is a set of ordered pairs. The relation may be specified by an equation, a rule, or a table. The set of the first components of the ordered pairs is called the...

Functions and Variations -Pre Calculus- Raye Rhouieze B. Miranda Functions  Relations A relation is a set of ordered pairs. The relation may be specified by an equation, a rule, or a table. The set of the first components of the ordered pairs is called the domain of the relation. The set of the second components is called the range of the relation. In this chapter, we shall consider only relations that have sets of real numbers for their domain and range.  Functions A function is a relation such that each element in the domain is paired with exactly one element in the range.  Functions Often functions and relations are stated as equations. When the domain is not stated, we determine the largest subset of the real numbers for which the equation is defined, and that is the domain. Once the domain has been determined, we determine the range by finding the value of the equation for each value of the domain. The variable associated with the domain is called an independent variable and the variable associated with the range is called the dependent variable. In equations with the variables x and y, we generally assume that x is the independent variable and that y is the dependent variable. Functions Functions  Function Notation The notation y = f (x), read "y equals f of x," is used to designate that y is a function of x. With this notation f(a) represents the value of; the dependent variable y when x = a (provided that there is a value). Thus y = x² − 5x + 2 may be written f (x) = x² − 5x + 2. Then f (2), i.e., the value of f(x) or y when x = 2, is f (2) = 2² − 5(2) + 2 = −4. Similarly, f (−1) = (−1)² − 5(−1) + 2 = 8. Any letter may be used in the function notation; thus g(x), h(x), F(x), etc., may represent functions of x.  Rectangular Coordinate System A rectangular coordinate system is used to give a picture of the relationship between two variables. Consider two mutually perpendicular lines X′X and Y′Y intersecting in the point O, as shown  Rectangular Coordinate System Given a point P in this xy plane, drop perpendiculars from P to the x and y axes. The values of x and y at the points where these perpendiculars meet the x and y axes determine respectively the x coordinate (or abscissa) of the point and the y coordinate (or ordinate) of the point P. These coordinates are indicated by the symbol (x, y). Conversely, given the coordinates of a point, we may locate or plot the point in the xy plane. For example, the point P in the figure has coordinates (3, 2); the point having coordinates (−2, −3) is Q. The graph of a function y = f (x) is the set of all points (x, y) satisfied by the equation y = f (x).  Function of Two Variables The variable z is said to be a function of the variables x and y if there exists a relation such that to each pair of values of x and y there corresponds one or more values of z. Here x and y are independent variables and z is the dependent variable. The function notation used in this case is z = f (x, y): read "z equals f of x and y." Then f (a, b) denotes the value of z when x = a and y = b, provided the function is defined for these values. Thus if f (x, y) = x³ + xy² − 2x, then f (2, 3) = (2³) + (2 · 3²) − (2 · 3) = 20. In like manner we may define functions of more than two independent variables.  Graphs of Functions In each case, for any value of x, there is only one value for y. Contrast this with the graphs  Graphs of NOT Functions In each of the relations, a single value of x is associated with two or more values of y. These relations are not functions.  Symmetry When the left half of a graph is a mirror image of the right half, we say the graph is symmetric with respect to the y axis. This symmetry occurs because for any x value, both x and −x result in the same y value, that is f (x) = f (−x). The equation may or may not be a function for y in terms of x.  Symmetry Some graphs have a bottom half that is the mirror image of the top half, and we say these graphs are symmetric with respect to the x axis. Symmetry with respect to the x axis results when for each y, both y and −y result in the same x value. In these cases, you do not have a function for y in terms of x.  Shifts The graph of y = f (x) is shifted upward by adding a positive constant to each y- value in the graph. It is shifted downward by adding a negative constant to each y-value in the graph of y = f (x). Thus, the graph of y = f (x) + b differs from the graph of y = f (x) by a vertical shift of |b| units. The shift is up if b > 0 and the shift is down if b < 0. Shifts  Shifts How do the graphs of y = (x + 1)² and y = (x − 2)² differ from the graph of y = x²? The graph of y = x² is shifted 1 unit to the left to yield the graph of y = (x + 1)² since x + 1 = x − (−1) The graph of y = x² is shifted 2 units to the right to yield the graph of y = (x − 2)² Shifts Seatwork #2 1. Given y = 2x − 1, obtain the values of y corresponding to x = −3, −2, −1, 0, 1, 2, 3 and plot the points (x, y) thus obtained. 2. Obtain the graph of the function defined by y = x2 − 2x − 8 or f (x) = x2 − 2x − 8. 3. Graph the function defined by y = 3 − 2x − x2. 4. In which of these equations is y a function of x? a. y = 3x³ b. y² = x c. xy = 1 d. y = 2x + 5 e. y=√4x f. y² = 8x -The End-