Lesson: ____ Section: 1.1 Functions & The idea of a function: At Taco Bell, the amount of money Change we spend is a function of the number of tacos we order. The amount of gas we burn is a function of the number of miles we drive. The word function expresses the idea that knowledge of one fact tells us another. e.g. If we know the radius of a circle, then circumference is determined. C is a function of r. If the number of eggs is a function of the number of chickens what does that mean? We think of E as a function of C and we call this function f, so we represent this relationship with E = f(C) Definition of a function: A rule that takes certain numbers as inputs and assigns to each exactly one definite output number. The set of all inputs is called the domain of the function. The set of resulting outputs is called the range.

The domain of a function can be explicitly stated or simply implied. Sometimes we choose to restrict the domain. For example, in the chicken problem from before, it does not make sense to have a negative number of chickens, so we restrict the domain to values 0. Function? 2 1 2 1 1 0 3 2

3 2 3 -2 5 3 5 3 9 7 Independent vs. Dependent Which variable is independent vs. dependent? Sometimes this

is obvious, sometimes its up to us depending on our point of view. Previously, we used the number of chickens to determine the number of eggs E = f(C), but we could use eggs to find chickens as well C = g(E). If each output is associated with only one input and vice-versa, we call this relationship a one-to-one function (the input & output are married). The significance of this is that E is a function of C, and C is a function of E, so we can go in either direction easily without any ambiguity. This allows us to define a function as well as an inverse for that function. Note that some quantities are discrete (only certain values e.g. dates) while others are continuous, which means they can be any number. (e.g. time) The Rule Of Four A relationship between quantities can be represented in many ways. The four most common representations are verbal, numerical, graphical, & Graphs

The Ru Of Fou Tables or Charts Equatio ns Words Example: Equations y=kx The Ru Of Fou Graphs All four of these

represent the same relationship! We want to shift effortlessly between them. Words y is proportional to x Tables or Charts x y 1 3 2 6 5

15 If we say The number of chickens is a function of the number of eggs. We can get our heads around the meaning of this statement using the rule of four! The key is to determine which variable is the input variable and which is the output. Graph: c e Equation: Numerical: n o i t c n

Fu ation t o N Domain x input Function f output Range f(x) x f f(x) o i

t c n Fu o N n n o i tat input name of the function output of function f at input x What does it mean??? f(2) = 9 f(2x) = 16 f(x) f(-x) = -f(x) g(x) = f(x) + 7

If h(x) = x2 + 4x, find h(3x) See Interwrite file for domain & difference quotients add piecewise Intervals are regions between values (inter vals... between values) Graphical Notation for Intervals: 5 10 Algebraic Notation for Intervals (using Inequalities): We say that 5 is an open endpoint (there is no first value in the interval) Interval Notation: 10 is a closed endpoint (10 is the last value in the interval) Ex. ( - 5, 2] [ - 2, ) = [-2, 2] of the intervals

Ex. ( - 5, 2] [ - 2, ) = (-5,) of the intervals 1. Is this a function ? 2. What is the domain? Range? 3. Over what intervals is it Increasing, Decreasing, Constant 4. Where is the function > 0, < 0 ?

5. Where is the slope incr. or decr.? 6. Are there any maxima or minima? Are they relative(local) or global? 7. Is the function even, odd, or neither? Why? ( ) = () ( ) = () Analyzing the Graph of a Function using Interval Notation When a graph is concave up, its slope is increasing.

When concave down, its slope is decreasing Difference Quotients A quotient of 2 differences ( 2 ) ( 1) 2 1 Think: Linear Functions = ( )=+ What are the constants in this generic equation? Variables? Formulas like this in which the constants can take on various values give us a family of functions. These constants (called the parameters) alter the parent function of the family in predictable ways. The Point-Slope Form of a Linear Equation withslope passingthrought he point( 1 , 1 )

= 1 + ( 1) + () Our favorite form for Calculus! + This makes sense! is my initial value and is how much the function has changed since then. = 4+7 ( 5) No need to simplify If y is directly proportional to x, then y = k x If y is inversely proportional to x, then y = k ( 1/x ) k is called the constant of proportionality

Ex. P = f(g) = 5g What happens to P if I double the input g? Ex. M = f( t ) = 10 ( 1/t ) What happens to M if I double the input t? Solving Polynomial Inequalities Steps: 1. Set inequality to zero 2. Factor if possible 3. Set each factor = 0 to obtain the roots 4. Place the critical number on a number line 5. Test a value in each interval to determine if the function is positive or negative within that interval (These are called test intervals) 6. Write your solution using interval notation Ex. We need to test to see on which intervals this product is negative!

+ < ( + )( )< ( + ) = ( )= = = ( + ) + + ( + ) ( ) The solution interval is Now, confirm this algebraic solution by graphing the function and looking to see when the outputs are negative! Graph the functions on your calculator and sketch the graphs on the same set of axes on your paper. ( )= + g

1. Over what intervals is f(x)0 2. Over what intervals is g(x)>3 3. Over what intervals is f(x) g(x) 4. Locate all extrema and classify them as relative/absolute 5. Over what intervals is f(x) increasing? 6. Over what intervals is g(x) increasing? 7. Over what intervals is the slope of g(x) increasing? 8. Is f(x) even, odd, or neither?