MATH 150 Pre-Calculus Fall, 2014, WEEK 6 JoungDong Kim Week 6: 4A, 4B, 4C, 4D Chapter 4. Functions Chapter 4A. Introduction to Functions Definitions A function is a relation in which no two different ordered pairs have the same first element. • A function can also be defined as a rule that assigns exactly one element in set B to each element • in set A. Set A, the set of all first coordinates, is called the domain of the function. • Set B, the set of all second coordinates, is called the range. • Note. There is only one output (y-value) for each input (x-value). 1 Ex1) Function or Not? a) (2, 3), ( 1, 3), (5, 3), (8, 3) { − } b) (1, 5), (2, 7), (3, 9), (1, 4) { } Notation. y = f(x) : x is input, f(x) or y is output. Evaluating Functions To evaluate the function f, we find the output for a given input. Ex2) For f(x)=2x2 x + 1, find f( 3), f(a + h), and f( x). − − − 2 f(x + h) f(x) Ex3) Evaluate − , h = 0 for the function f(x)=2x2 x + 1. h 6 − f(x + h) f(x) Ex4) Evaluate − , h = 0 for the function f(x)= x2 x 1. h 6 − − − DIY. 3 Domain of a Function The domain of a function is the set of all inputs for the function. In Sum, The domain of most algebraic functions is the set of all real numbers EXCEPT: 1. any real numbers that cause the denominator of a function to be 0. 2. any real numbers that cause us to take the square root (or any radical with an even index) of a negative number. and more... 4 Ex5) Find the domain of each function. 1 a) f(x)= x2 x − b) g(x)= √2x 3 − t c) h(t)= √t +1 5 Ex6) Find the domain and range of each function. a) f(x)= x2 5x +6 − b) g(x)= √x2 5x +6 − 6 √x +2 Ex7) Find the domain of f(x)= . x 1 − 7 Applying Functions Ex8) If x and y are the lengths of the adjacent sides of a rectangle whose parameter is p, write y as a function of x. Ex9) An open box is constructed from a 12-inch square piece of cardboard by cutting squares of equal length from each corner and turning up the sides. Write a function for the volume of the cardboard box in terms of the length of the cutout square. Restrict the domain appropriately. 8 Chapter 4B - Graphs of Functions Definition If f is a function with domain A, then the graph of f is the set of ordered pairs (x, f(x)), where x A. ∈ In other words, the graph of f is the set of all points (x, y) such that y = f(x). Note. y = mx + b: linear function, straight line. Ex10) Graph the function f(x)= √x + 3, and find the domain, range, x-intercept, and y-intercept. 9 Find domain and range from Graphs; To find the domain of f, vertical line scanning the x-axis. To find the range of f, horizontal line scanning the y-axis. 10 Ex11) Find x- and y-intercepts of the graph of f(x)= x3 9x. − 11 Vertical Line Test A set of points in the coordinate plane represents a function if and only if no vertical line intersects the graph in more than one point. 12 Catalog of Basic Functions 1. Constant Function f(x)= c, c : real constant 2. Identity Function f(x)= x 3. Squaring Function f(x)= x2 13 4. Cubing Function f(x)= x3 5. Square Root Function f(x)= √x 6. Absolute Value Function f(x)= x | | 7. Reciprocal 1 f(x)= x 14 Piecewise Functions x, x< 0 Ex12) Graph f(x)= by plotting points. x2, x 0 ≥ 15 Ex13) Graph the piecewise function x, if x 0 f(x)= ≥ x, if x< 0 − 16 Ex14) Graph the piecewise function 2 x if x> 1 f(x)= 3 if x =1 x + 1 if x< 1 − 17 Increasing, Decreasing, and Constant Definition. A function f is increasing on an interval I if and only if for every x1 < x2 I, ∈ f(x1) < f(x2). Definition. A function f is decreasing on an interval I if and only if for every x1 < x2 I, ∈ f(x1) > f(x2). Definition. A function f is constant on an interval I if and only if for every x1, x2 I, f(x1)= ∈ f(x2). 18 Ex15) Use the graph to find the following for each function. Domain • Range • Intercepts • Intervals where f is increasing, decreasing, and constant • 19 Chapter 4C. Transformations of Functions Vertical and Horizontal Shifting Adding a constant to a function shifts its graph vertically: upward if the constant is positive and downward if it is negative. Ex16) Use the graph of f(x)= x2 to sketch the graph of each function. a) g(x)= x2 +3 b) h(x)= x2 2 − Vertical Shifts of Graphs Suppose c> 0. To graph y = f(x)+ c, shift the graph of y = f(x) upward c units. To graph y = f(x) c, shift the graph of y = f(x) downward c units. − 20 Horizontal Shifts of Graphs Suppose c> 0. To graph y = f(x c). shift the graph of y = f(x) to the right c units. − To graph y = f(x + c). shift the graph of y = f(x) to the left c units. Ex17) Use the graph of f(x)= x2 to sketch the graph of each function a) g(x)=(x + 4)2 b) h(x)=(x 2)2 − 21 Ex18) Sketch the graph of f(x)= √x 3 + 4. − 22 Ex19) Sketch the graph of f(x)=(x 1)2 + 2. − 23 Reflecting Graphs To graph y = f(x), reflect the graph of y = f(x) in the x-axis. To graph y = f−( x), reflect the graph of y = f(x) in the y-axis. − Ex20) Sketch the graph of each function. a) f(x)= x2 b) g(x)= √ x − − 24 Vertical Stretching and Shrinking The y-coordinate of y = cf(x) at x is the same as the corresponding y-coordinate of y = f(x) multiplied by c. Multiplying the y-coordinates by c has the effect of vertically stretching or shrinking the graph by a factor of c. Vertical Stretching and Shrinking of Graphs. To graph y = cf(x); If c> 1, stretch the graph of y = f(x) vertically by a factor of c. If 0 <c< 1, shrink the graph of y = f(x) vertically by a factor of c. Ex21) Use the graph of f(x)= x2 to sketch the graph of each function. 1 a) g(x)=3x2 b) h(x)= x2 3 25 Ex22) Sketch the graph of the function f(x)=1 2(x 3)2 − − 26 Ex23) Sketch the graph of the function f(x)= x3 2 − − 27 1 Ex24) Sketch the graph of the function f(x)= +3 x 2 − 28 Ex25) Sketch the graph of the function f(x)= 3 x +1 2 − | | − 29 Even and Odd Functions Let f be a function. f is even if f( x)= f(x) for all x in the domain of f. − f is odd if f( x)= f(x) for all x in the domain of f. − − Note. The graph of an even function is symmetric with respect to the y-axis, The graph of an odd function is symmetric with respect to the origin. 30 Ex26) Determine whether the functions are even, odd, or neither even nor odd. a) f(x)= x5 + x b) g(x)=5x4 31 Chapter 4D. Maximum/Minimum Function Values. Quadratic Functions A quadratic function is a function f of the form f(x)= ax2 + bx + c where a, b, and c are real numbers and a = 0. The graph of a quadratic function is parabola. 6 Standard form of a Quadratic Function A quadratic function f(x)= ax2 + bx + c can be expressed in the standard form f(x)= a(x h)2 + k − by completing the square. The graph of f is a parabola with vertex (h, k): the parabola opens upward if a> 0 or downward if a< 0. 32 Ex27) Let f(x)=2x2 12x + 23, express f in standard form and sketch the graph of f. − 33 Maximum or Minimum Value of a Quadratic Function Let f be a quadratic function with standard form f(x)= a(x h)2 + k. The maximum or minimum value of f occurs at x = h. − If a> 0, then the minimum value of f is f(h)= k. If a< 0, then the maximum value of f is f(h)= k. Ex28) Find the maximum or minimum value of f(x)=5x2 30x + 49. − 34 Ex29) Write f(x) = 2x2 + 10x 7 in standard form. Draw the graph and find vertex, axis − − of symmetry, domain, range, the maximum or minimum value, and the intervals of increasing and decreasing. 35 If we interested only in finding the maximum or minimum value, then a formula is available for doing so. This formula is obtained by completing the square for the general quadratic function as follows; 2 This equation is in standard form with h = b and k = b + c. Since the maximum or minimum − 2a − 4a value occurs at x = h, we have the following result Maximum or Minimum value of a Quadratic Function. The maximum or minimum value of a quadratic function f(x)= ax2 + bx + c occurs at b x = .