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Transcript
SECTION 1.4 Function Notation FUNCTION AS A PROCESS The domain of a function is a set of inputs and The range is a set of outputs A function f is a process by which each input x is matched with only one output y y = f(x) is the output which results by applying the process f to the input x INDEPENDENT AND DEPENDENT VARIABLES The value of y is completely dependent on the choice of x x is often called the independent variable or argument of f y is often called the dependent variable ALGEBRAIC FORMULA OF FUNCTION The process of a function f is usually described using an algebraic formula Example: a function f takes a real number and performs the following two steps 1. 2. multiply by 3 add 4 If 5 is our input, in step 1 we multiply by 3 to get (5)(3) = 15. In step 2, we add 4 to our result from step 1 which yields 15 + 4 = 19 Using function notation, we would write f(5) = 19 to indicate that the result of applying the process f to the input 5 gives the output 19. In general, if we use x for the input, applying step 1 produces 3x. Following with step 2 produces f(x) = 3x + 4 as our final output. EXAMPLE Suppose a function g is described by applying the following steps, in sequence 1. 2. add 4 multiply by 3 Determine g(5) and find an expression for g(x) EXAMPLE For f(x) = -x2 + 3x + 4, find and simplify 1. 2. 3. f(-1), f(0), f(2) f(2x), 2 f(x) f(x + 2), f(x) + 2, f(x) + f(2) IMPLIED DOMAIN Suppose we wish to find r(3) for r(x) =2x/(x2 – 9) The number 3 is not an allowable input to the function r; in other words, 3 is not in the domain of r When a formula for a function is given, we assume the function is valid for all real numbers which make arithmetic sense when substituted into the formula. This set of numbers is often called the implied domain of the function. EXAMPLE Find the domain of the following functions EXAMPLE This example shows how a function can be used to model real-world phenomena The height h in feet of a model rocket above the ground t seconds after lift off is given by 5t 2 100t , if h(t ) 0, if 0 t 20 t 20 Find and interpret h(10) and h(60) This type of function is called a piecewise function