Section 2.1 and 2.2. I. Relation - A correspondence between two variables that can be used to describe how one variables affects the other variable.

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1 1 Section 2.1 and 2.2 In algebra, we often let letters represent a variable. This is because the letters represent numbers that can vary in value. Ex: Scientists found that the wolf population in Yellowstone National Park has increased from What are the two variables? I. Relation - A correspondence between two variables that can be used to describe how one variables affects the other variable. There are THREE ways to describe the relationship between two variables. 1. A set of ordered pairs, often written in a Table Example: The ordered pairs below represent years since 1999 and wolf population. {(0, 72), (1, 119), (2, 132), (3, 148), (4, 174), (5, 169)} Write the ordered pairs in the table. 2. Graphs Example: Plot the points from the table on a graph. 3. Equations Equation A relationship between 2 variables x and y involving an = sign. Ex: y x 2 4 x y e y 5x 2

2 2 II. Definitions Domain: The set of all first components (x-values) of the ordered pairs. Range: The set of all second components (y-values) of the ordered pairs. X-Intercept: The y-value when x = 0. Y-Intercept: The x-value when y = 0. Example: Write the domain and range for {(0, 72), (1, 119), (2, 132), (3, 148), (4, 174), (5,169)}. Example: Use the graph to answer the following questions. a) Write the Domain using set-builder notation and interval notation. b) Write the Range using set-builder notation and interval notation. c) What is (are) the x-intercept(s)? d) What is (are) the y-intercept(s)?

3 3 III. Function A relation such that for each x-value in the domain, there is exactly one corresponding y-value in the range. Independent Variable The input of the function, often represented by x. Dependent Variable The output of the function, often represented by y. Notation: 1) If a variable y is a function of variable x, we write y = f(x) Dependent var = f (independent var) Output = f (input) Where f = function name y = dependent variable x = independent variable 2) The ordered pair (x, y) can now be written as (x, f(x)). This still represents (independent var, dependent var) (input, output) Comment: All functions are equations, but not all equations are functions. A. Functions and Tables: Two different x-values may or may not have the same y-value. Two same x-values must have the same y value. Example: Identify whether each relation is a function. a) {(1, 6), (2, 7), (3, 8)} b) c) f(1) = y, y =? f(x) = 7, x =? f(x) = 6, x =? f(2) = y, y =?

4 4 B. Functions and Graphs: Vertical Line Test (VLT) - Each value of x on the horizontal axis must equal only one value on the y-axis.

5 5 Increasing Function: f(x) increases as x increases (reading the graph left to right). Decreasing Function: f(x) decreases as x increases (reading the graph left to right). Relative Maximum: Lowest y-value on the graph. Relative Minimum: highest y-value on the graph. Even Function: Odd Function: f(-x) = f(x) The graph is symmetric about the y-axis f(-x) = -f(x) The graph is symmetric about the origin Concave Up: The graph bends up. Concave Down: The graph bends down. Where is f(x) < 0, f(x) > 0, f(x) = 0? How many times does the line y = 2 intersect the graph? Where does f(x) = -5? List the x-values. Find 3f(1).

6 6 C. Functions and Equations To determine if an equation is a function: Method I: 1. Solve the equation for y. 2. If only one value of y can be obtained for an x, then the equation is a function. 3. If more than one value of y can be obtained for an x, then the equation is not a function. Method II: 1. Graph the equation and run the VLT. To graph, pick out some x-values and find the y-values. Write them as an ordered (x, y) pair. Plot the points. To write an equation in function form: 1. Solve the equation for y. 2. Substitute y = f(x) into the solved equation. Example: Determine whether each equation defines y as a function of x. If so, write the equation in function form. a) 2 4x y b) 3 x y 27 c) xy 2y 1 We can evaluate a function at a given value of the independent variable.

7 7 Example: Given f( x) 1 x 2 and 2 g( x) 2x 3x 1, evaluate: a) f(0) b) f(2) c) g(3a) d) g(x+h) Difference Quotient of function f: f ( x h) f ( x) h, h 0 Example: Find the difference quotient for g(x).

8 8 IV. Piecewise Functions- A function defined by two (or more) equations over a specified domain. Example: The absolute value function f ( x) x Evaluate the piecewise function at f(-5). Graph the piecewise function. Determine the range of the function. Example: a) Where is f(x)<0? b) Where is f(x) 0? c) Where is f(x) increasing? d) Where is f(x) decreasing?

9 9 Example: Graph the piecewise function. Determine the range of the function. Example: Evaluate the piecewise function at the given values of the independent variable.

10 10 Example WARNING: WE WILL ANSWER SOME OF THE QUESTIONS BELOW NOT ALL QUESTIONS DEPENDING ON CLASS TIME!

11 11 Section 2.3 and 2.4 Average Rate of Change (ARC) = Change in y Change in x = = y x f ( x1) f ( x2) x x 1 2 where f is a function, and ( x1, f ( x 1)) and ( x2, f ( x 2)) are distinct points. Example: Let X = year since 1994, and Y = cocaine related emergency room episodes (in thous.) Data Set One Year (X) Emerg. Room (Y) ARC Some functions have a different ARC between each successive interval. Data Set Two Year (X) Emerg. Room (Y) ARC Linear functions have a constant ARC on every interval. The graph of a linear function is a straight line.

12 12 Another word of Average Rate of Change (ARC) is SLOPE SLOPE = m = y x y x or y x y x for points and ( x1, y 1) and ( x2, y 2) Ex: What s the slope of the line over the interval [-5, -2] on the graph below? Interpreting Slope: Positive Slope (m > 0): Y increases as X increases. Negative Slope (m < 0): Y decreases as X increases. Zero Slope (m = 0): Y remains constant as X increases. Line is horizontal. (Points on the line are collinear). Undefined Slope: Line is vertical.

13 13 I. Slope-Intercept Form A function y = f(x) is a linear function written in slope-intercept form if it can be represented in the form: y = mx + b or y = b + mx 1. m = slope = y x y x a. The larger m, the steeper the graph. b. Controls how fast the line rises and falls. Comments: 1. m determines the steepness of the line. 2. The scale used on a graph affects how steep the slope appears. Always verify which graph has a steeper slope by calculation, not appearance. 3. Watch your units when calculating slope. Ex: Episodes per year Ex: Dollars per gallon

14 14 2. b = vertical intercept / y-intercept a. Where the line crosses the y-axis. b. Value of y when x = 0. c. The starting point (0, y) d. To find, plug in 0 for x and solve for y!!

15 15 Example: A new $50,000 Dodge Ram Outdoorsman depreciates in value (V) by $5000 per year (t). To graph a linear function: 1. Plot the point (0, b). 2. Use m to obtain successive points. Ex. Plot g(x) = 2 3 x Ex: Plot 3x + y 5 = 0.

16 16 To write the slope-intercept form of an equation given two points: 1. Find m 2. Plug in m, x 1, and y 1 3. Solve for b. 4. Rewrite the equation using x and y as variables and values of m and b. Ex: Write the equation of the line passing through the points (2, 1) and (3, 4). Express your answer in slope-intercept form.

17 17 II. Special Types of Lines Horizontal Line y = b m = 0 Passes through (0, b) Ex: Consider f(x) = 4. a) Graph b) What is f(0)? c) What is the slope? Vertical Line x = a m = undefined Passes through (a, 0)

18 18 Parallel Lines Have the same slope. y m x b and y2 m2 x2 b 2 are parallel, then m1 m 2 If Ex: Write the equation of the line parallel to 3x 2y 5 = 0 and passing through the point (-1, 3). Express your answer in slope-intercept form. Perpendicular Lines Slopes are negative reciprocals. If y1 m1 x1 b 1 and y2 m2 x2 b 2 are perpendicular, then m1 1 m 2 Ex: Write the equation of the line perpendicular to x-2y-3 = 0 and passing through the point (4, -7). Express your answer in slope-intercept form. Ex: Write the equation of the line L in slope-intercept form.

19 19 III. Point Slope Form A function y = f(x) is a linear function written in point slope form if it can be represented in the form: y y1 m x x 1 Ex: Write the equation of the line with slope = 6, passing through (-2, 5). Express your answer in point slope form. Ex: Express your answer to the previous question in slope intercept form. To graph: 1. Plot the point ( x 1, y 1 ) 2. Use m to obtain a second point. Ex: Graph the linear function y 2 = 2 5 (x+1). Also, rewrite the equation in slope intercept form. Then graph the equation using the slope intercept form.

20 20 IV. Standard Form A function y = f(x) is a linear function written in standard form if it can be represented in the form: Ax By C Where A, B, and C are constants. Example: Convert the equation y 2 = 2 (x+1) from point slope form to standard form. 5

21 21 To graph an equation written in standard form: 1. Calculate the y- intercept Plug x = 0 into the equation. Solve for y. obtain the point (0, y) 2. Calculate the x- intercept Plug y = 0 into the equation. Solve for x. obtain the point (x, 0) 3. Plot the points (0, y) and (x, 0). Connect these points with a line. Example: Write the equation 6x-9y 18 = 0 in standard form. Then use intercepts to graph.

22 22 Section 2.5 Graphs of Parent Functions that YOU MUST KNOW.

23 23 Properties of the Parent Functions THAT YOU MUST KNOW:

24 24 7. The Cube-Root Function Y = f(x) = Properties: 1. Domain: x (, ) 2. Range: y (, ) 1 3 x = 3 x 3. Y-intercept: (0, 0); x-intercept: (0, 0) 4. Increasing over (, ) 5. Symmetry: About the origin (odd function)

25 25 Transformations Let f (x) be a function and c be a positive real number. 1. Horizontal Shift f(x+c) shifts the graph LEFT by c units. f(x-c) shifts the graph RIGHT by c units. 2. Shrink/Stretch cf(x) multiplies f(x) / the y-coordinates by the constant c. c > 1, the graph of f(x) is stretched. 0 < c < 1, the graph of f(x) is compressed/shrinks.

26 26 3. Reflection of f(x) across the x-axis Multiply f(x) by -1: -f(x). 4. Reflection of f(x) across the y-axis. Multiply x by -1: f(-x) 5. Vertical Shift f(x) + c shifts the graph UP by c units. f(x) - c shifts the graph DOWN by c units.

27 27 Comment: Some functions involving more than one transformation. All transformations must be applied in the above order for this class. In pre-calculus you will learn other possible orders that the transformations can be applied. Example: Let f(x) = 2 x. Use transformations to graph g(x) = 2 2( x 2) 1. Example: Let f(x) = x. Use transformations to graph g(x) = 2 x 1 1.

28 28 Example: Let f(x) = x. Use transformations to graph g(x) = 3 x 3. Example: Let f(x) = 3 x. Use transformations to graph g(x) = 3 ( x 2) 1.

29 29 Example: Let f(x) = 2 x. Use transformations to graph g(x) = 2 x 2. Example: Let f(x) = 3 x. Use transformations to graph g(x) = 3 x 2.

30 30 More practice problems with solutions (probably won t have time to cover these in class).

31 31 Solutions

32 32 Section 2.6 There are two Operations on Functions that we will study: 1. Combination of functions 2. Composition of functions Combination of Functions I. Let f(x) and g(x) be any two functions. Then: 1. f(x) + g(x) = (f + g)(x) is the sum of the two functions. 2. f(x) - g(x) = (f - g)(x) is the difference of the two functions. 3. f(x) g(x) = (f g)(x) is the product of the two functions. 4. f( x) gx ( ) f = ( x) g is the quotient of the two functions. Restriction: g(x) 0. Example: Let f(x) = 2 x +1, and g(x) = 3x+5. Find each of the following: a) (f + g)(1) b) (f - g)(1) c) (f g)(1) d) (1) f g

33 33 II. When operating on functions, we must be sure to state the domain in either: 1. Set Builder Notation: {x x } 2. Interval Notation: () or [] or (] or [) Exclude numbers from the domain of a function that cause: 1. DIVISION BY ZERO 2. THE EVEN ROOT OF A NEGATIVE NUMBER Example: State the domain of each function below in set builder notation and interval notation. a) 1 4 x 1 2 b) x x 2 5

34 34 III. The domain of combined functions consists only the numbers in the domain of BOTH f individual functions. The only tricky case is ( x). g Be sure to find the domain function BEFORE simplifying. Example: Find each combined function and state its domain. 3x 1 a) f( x) 2 x 25 gx ( ) 2x 4 2 x 25 Find (f - g)(x) and f( x) gx ( ) f = ( x) g b) f ( x) x 5 g( x) 5 x Find (f g)(x) and ( ) f x g

35 35 Composition of Functions I. If f(x) and g(x) are two functions, then the composite function is given by f ( x) g( x ) = ( f g)( x ) = f(g(x)) Example: Let f ( x) 7x 1 and a) ( f g )(2) 2 g( x) 2x 9. Find b) ( f g)( x ) c) ( g f )( x ) at x = 2. II. The domain of composite functions If x is not in the domain of g, then it is not in the domain of f(g(x)). Consider x = #. If the value of g(#) is not in the domain of f(x), the it is not in the domain of f(g(x)). Example Find the composite functions f(g(x)) and state the domain when: f(x) = 2 x 3 and gx ( ) 1 x

36 36 Example: Use the table below to answer the following questions. x f(x) g(x) a) f g (1) b) f(g(0)) c) f(f(x)) at x = 2 d) g(f(1)) Example: Use the graph below to answer each of the questions. f(x) g(x) a) g f (2) b) f(g(1)) c) g(f(1))

37 37 Section 2.7 Inverse Functions Functions f(x) and g(x) are inverse functions of one another if: f g ( x) x and g f ( x) x f(g(x)) = x and g(f(x)) = x Example: Verify whether the functions below are inverses. f(x) = 2x + 1 g(x) = x 1 2 Example: Show that the functions 3 f ( x) 4x 5 and 3 x 5 g ( x) are inverse functions. 4 Notation: If f(x) is the inverse of g(x), we write f(x) = g 1 ( x ). If g(x) is the inverse of f(x), we write g(x) = f 1 ( x ).

38 38 Steps to find the inverse function. Idea: y = f(x) x = f 1 ( y ) 1. Replace f(x) by y. 2. Switch x and y. 3. Solve for y. 4. Write y = f 1 ( x ) = 5. State the domain of f 1 ( x ) these will be x-values (RANGE OF f(x)). State the range of f 1 ( x ) these will be y-values (DOMAIN OF f(x)). Example: Find the inverse of each function below. Use interval notation to state the domain and range of f(x) and f 1 ( x ). a) f 3 ( x) x 4 b) f ( x) 9 x

39 39 Rule: Functions that have an inverse must be one-to-one. Algebraically, a function is one-to-one if whenever x 1 x2 then f ( x1 ) f ( x2). Graphically, a function is one-to-one if it passes the Horizontal Line Test. HLT: Nor horizontal line can cross the graph twice. Example: Determine whether the following functions have an inverse.

40 40 To compare the graphs of a function and its inverse, know: If point (a, b) is on the graph of f(x), then the point (b, a) is on the graph of f 1 ( x ). The graph of f 1 ( x ) is the reflection of f(x) about the line y = x. Example: Example: Let function f(x) be the blue line. Graph the inverse of f(x).

41 41 Example (time permitting) a) Write a function to model the information in the first frame of the cartoon. b) Find the inverse of this function. The inverse expresses the number of cricket chirps per minute as a function of Fahrenheit temperature.