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1.2 Functions • Determine whether relations between two variables represent functions • Use function notation and evaluate functions • Find the domains of functions • Use functions to model and solve real-life problems • Evaluate difference quotients Definition of a Function: A function is a relation in which each element of the domain (the set of x-values, or input) is mapped to one and only one element of the range (the set of y-values, or output). Function Not a Function One-to-one Function A Function can be represented several ways: • Verbally – by a sentence that states how the input is related to the output. • Numerically – in the form of a table or a list of ordered pairs. • Graphically – a set of points graphed on the x-y coordinate plane. • Algebraically – by an equation in two variables. Example 1 Input x 2 2 3 4 5 Output y 11 10 8 5 1 Example 2 Which of the equations represents y as a function of x? a. x 2 y 1 b. x y 2 1 Example 3 Let g( x) x g(2)= g(t)= g(x+2)= 2 4x 1 Example 4 : Evaluate the piecewise function when x=-1 and x=0. {x 1, x 0 {x 1, x 0 2 Example 5 : Find the domain of each function a. f: {(-3,0),(-1,4),(0,2),(2,2),(4,-1)} b. 3x 2 4 x 5 c. 1 h( x ) x 5 d. 4 V r3 3 e. k ( x) 4 3x Example 6 Use a graphing calculator to find the domain and range of the function f ( x) 9 x 2 Example 7 The number N (in millions) of cellular phone subscribers in the United States increased in a linear pattern from 1995 to 1997, as shown on p.22. Then, in 1998, the number of subscribers took a jump, and until 2001, increased in a different linear pattern. These two patterns can be 10.75t 201 . ,5 t 7 approximated by the function N (t ) { where t represents the year, with 2011 . t 92.8,8 t 11 t=5 corresponding to 1995. Use this function to approximate the number of cellular phone subscribers for each year from 1995 to 2001. Example 8 A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45 degrees. The path of the baseball is given by the function f ( x) .0032 x 2 x 3 where y and x are measured in feet. Will the baseball clear a 10 foot fence located 300 feet from home plate? Student Example A baseball is hit at a point 4 feet above the ground at a velocity of 120 feet per second and an angle of 45 degrees. The path of the baseball is given by the function f ( x) .0038x 2 x 4 where y and x are measured in feet. Will the baseball clear an 8 foot fence located 350 feet from home plate? Example 9 For f ( x) x 2 4 x 7, find f ( x h) f ( x ) . h Student Example Evaluate f ( x) 2 3x x 2 for f(-3) f(x+1) f(x+h)-f(x)