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§ 0.2 Some Important Functions Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 78 Section Outline Linear Equations Applications of Linear Functions Piece-Wise Functions Quadratic Functions Polynomial Functions Rational Functions Power Functions Absolute Value Function Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 78 Linear Equations Equation Example y = mx + b (This is a linear function) x=a (This is not the graph of a function) Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 78 Linear Equations CONTINUED Equation Example y=b Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 78 Applications of Linear Functions EXAMPLE (Enzyme Kinetics) In biochemistry, such as in the study of enzyme kinetics, one encounters a linear function of the form f x K / V x 1/ V , where K and V are constants. (a) If f (x) = 0.2x + 50, find K and V so that f (x) may be written in the form, f x K / V x 1/ V. (b) Find the x-intercept and y-intercept of the line f x K / V x 1/ V in terms of K and V. SOLUTION (a) Since the number 50 in the equation f (x) = 0.2x + 50 is in place of the term 1/V (from the original function), we know the following. 50 = 1/V 50V = 1 V = 0.02 Explained above. Multiply both sides by V. Divide both sides by 50. Now that we know what V is, we can determine K. Since the number 0.2 in the equation f (x) = 0.2x + 50 is in place of K/V (from the original function), we know the following. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 78 Applications of Linear Functions CONTINUED 0.2 = K/V 0.2V = K 0.2(0.02) = K 0.004 = K Explained above. Multiply both sides by V. Replace V with 0.02. Multiply. Therefore, in the equation f (x) = 0.2x + 50, K = 0.004 and V = 0.02. (b) To find the x-intercept of the original function, replace f (x) with 0. f x K / V x 1/ V 0 K / V x 1/ V 1/ V K / V x This is the original function. Replace f (x) with 0. Solve for x by first subtracting 1/V from both sides. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 78 Applications of Linear Functions CONTINUED 1/ V V / K x 1 / K x Multiply both sides by V/K. Simplify. Therefore, the x-intercept is -1/K. To find the y-intercept of the original function, we recognize that this equation is in the form y = mx + b. Therefore we know that 1/V is the y-intercept. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 78 Piece-Wise Functions EXAMPLE 1 x for x 3 . 2 for x 3 Sketch the graph of the following function f x SOLUTION We graph the function f (x) = 1 + x only for those values of x that are less than or equal to 3. 6 4 2 0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -2 -4 -6 Notice that for all values of x greater than 3, there is no line. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 78 Piece-Wise Functions CONTINUED Now we graph the function f (x) = 4 only for those values of x that are greater than 3. 6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Notice that for all values of x less than or equal to 3, there is no line. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 78 Piece-Wise Functions CONTINUED Now we graph both functions on the same set of axes. -6 -5 -4 -3 -2 6 5 4 3 2 1 0 -1 -1 0 -2 -3 -4 -5 -6 1 2 3 4 5 6 Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 78 Quadratic Functions Definition Example Quadratic Function: A function of the form f x ax 2 bx c where a, b, and c are constants and a 0. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 78 Polynomial Functions Definition Example Polynomial Function: A function of the form f x an x n an 1 x n 1 a0 f x 17 x3 x 2 5 where n is a nonnegative integer and a0, a1, ...an are given numbers. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 78 Rational Functions Definition Rational Function: A function expressed as the quotient of two polynomials. Example 3x x 4 g x 2 5x x 1 Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 78 Power Functions Definition Example Power Function: A function of the form f x x 5.2 f x x r . Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 14 of 78 Absolute Value Function Definition Example Absolute Value Function: The function defined for all numbers x by f x x f x x , such that |x| is understood to be x if x is positive and –x if x is negative f 1 2 1 2 1 2 Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 15 of 78