P.5-P.6 Functions & Analyzing Graphs of Functions p.58-84 Objectives: Determine whether relations between two variables are functions. Use function notation and evaluate functions. Find the domains of functions. Use functions to model and solve real-life problems. Evaluate difference quotients. Functions Introduction Vocabulary Function Input Variable A special relation that assigns each element from one set with exactly one element of a different set. The variable in which you will plug numbers/algebraic expressions into the function. Output Variable The variable that will represent the answer achieved by plugging in the input variable. Function Notation f(x) reads f of x (Note: f(x) is not f times x. The f and x cannot be separated.) x is the input (what will be plugged into the formula) f is the formula/operation/expression; f(x) is the output achieved by sending x through the function/formula. f is traditionally used for functions. But, other letters may also be used. Function Representations 1. Verbally (a sentence that relates the input and output variables) 2. Numerically (tables or lists) 3. Graphically (ordered pairs are of the form (x, f(x))) 4. Algebraically (formula/equation) Discrete Function These functions include unconnected points. Tables, sets of ordered pairs, and scatter plots that are functions represent this type of function. Continuous Function These functions include connected points. Most formulas and equations that are functions tend to represent this type of function.
Independent Variable This is the input variable. Dependent Variable Domain Range Piecewise-Defined Function This is the output variable. The set of all values taken on by the input/independent variable. In other words, it is the set of all numbers that are put into the function. The set of all values taken on by the output/dependent variable. In other words, it is the set of all answers. A function defined by two or more equations over a specified domain. Implied Domain Explicit Domain Since f(x) is a function, it is implied that the domain and range are subsets of the real numbers. In other words, if it is known that you are working with a function, then it is assumed that the independent values must make the dependent values real numbers. A domain that is written with the function. Symmetry Tests X-Axis Symmetry: ---------------------------------------------------------------------------------------------------------------- Y-Axis Symmetry: --------------------------------------------------------------------------------------------------------------- Origin Symmetry:
The graphs of these functions are symmetric with respect to the y-axis. The graphs of these functions are symmetric with respect to the origin. X-Coordinate The directed distance from the y-axis. Y-Coordinate The directed distance from the x-axis. Same as f(x). Graph of a Relation A collection of ordered pairs. A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point. It is used to determine if a graph is the graph of a function. The x-values for which f(x) = 0. The same as real zeros. Local/Relative Minimum Local/Relative Maximum Local/Relative Extrema Absolute/Global Maximum The minimum value of the function for a localized interval. The maximum value of the function for a localized interval. The collection of localized minima and maxima. The maximum value of the entire function. Absolute/Global Minimum The minimum value of the entire function.
Rate of Change The slope of the line through two specific points on the graph of a function. Secant Line A line that intersects a curve in two or more points. Examples EX 1: Which of the following discrete relations are also functions? 1. {(3,6),(2,5),(-1,4),(3,7)} 2. {(2,2),(-1,1),(5,8),(3,2),(-1,1),(4,5)} 3. {(1,2),(2,2),(3,6),(4, ),(-1,4)} 4. {(1,2),(2,3),(3,4),(4,5)} EX 2: Evaluate the following. Given: f(x) = x 2 1. f(a) 2. f(1/a) 3. f(-a) 4. f(a) 5. f(a + b) 6. f(a) + f(b) 7. Domain = {1, 2, 3, 4} and Range = {5, 6, 7} 5. {(1,5), (2,6), (3,7)} 6. {(1,5), (2,5), (3, 5), (4, 5)} 7. {(1, 5), (2, 6), (3, 7), (4, 6), (2, 7)} EX 3: Determine if the following relations are also functions? 1. x = y 2 EX 4: Is h even or odd? h(x) = x 3 5 2. x + y 2 = 4 3. y + 1 = x 2 4. 5.
EX 5: Evaluate the following. Given: EX 6: Find the domain of the following functions. 1. {(1,2),(2,2),(3,2),(4,2)} 1. f(-2) = 2. 2. f(1) = 3. f(2) = 3. EX 7: Evaluate the difference quotient. If f(x) =2 + 5x x 2, find. EX 8: Find f(x). Given: f(-2) = 5 and f(6) = 3
EX 9: Test for each type of symmetry: y = 2 x x 1 EX 10: Find the following information for the following graph of a function, f(x). 1. Domain: 2. Range: 3. f(4) = 4. Zeros: 5. Increasing: 6. Decreasing: 7. Constant: 8. Relative Minimum: 9. Relative Maximum: 10. Absolute Maximum: 11. f(x) 0: