Functions: Review of Algebra and Trigonometry

Transcription

1 Sec. and. Functions: Review of Algebra and Trigonoetry A. Functions and Relations DEFN Relation: A set of ordered pairs. (,y) (doain, range) DEFN Function: A correspondence fro one set (the doain) to anther set (the range) such that each eleent in the doain corresponds to eactly one eleent in the range. Eaple: Deterine whether each of the following is an eaple of a function or not..),3, 5,3.),,4 3, 5, 4 3.) y 0 4.) y Vertical Line Test for Functions Vertical Line Test for Functions: If any vertical line intersects a graph ore than once, then the graph is not a function. Eaple: Deterine whether each of the following is a function or not by the Vertical Line Test. Desiré Taylor Math 4

2 B. Doain and Range of a Function DEFN Doain: Input -values (i.e. All of the values of that I ay plug into a function.) DEFN Range: Output y-values (i.e. All of the values of y that a function can attain) Function Notation: f() = y Eaple: Give the doain and range (in interval notation) for each of the following.) Doain: Doain: Doain: Range: Range: Range: Eaple: Give the doain for each of the following (in interval notation and as an inequality).) g 3.) f ) g 5.) h ln6 3 * The 3 functions for which we will ost frequently have doain restrictions (in this course) are: fractions (aka rational functions), radicals and logariths. Desiré Taylor Math 4

3 C. Linear Models Definition rise Slope = = = run y y y = Slope Slope-Intercept For Point-Slope For y y General For A B y C 0 y (Use when given two points to find slope) y y b (Use when given slope and y - intercept) (Use when given one point and slope) Horizontal Line Vertical Line y b (where b = constant) c (where c = constant) Parallel Lines Two lines are parallel if and only if they have the sae slope. For two lines y b and y b we have y Perpendicular lines y Two lines are perpendicular if and only if the product of their slope = -. y For two lines y b and y b we have y Eaples: Find the equation of the line:.) that passes through point (0, -3) with slope = -.) that passes through points (3, -) and (4, 5) Desiré Taylor Math 4 3

4 3.) that passes through point (0, 0) and is parallel to the line y ) that passes through point (, -4) and is perpendicular to the line y 3 5.) Find the slope and y-intercept of the line 9 3y 3 0 D. Classes of Functions. Power Functions For any real nuber, a function in the for f is called a Power Function. Polynoials Definition n n n A polynoial function is a function in the for f an an an a a0 a 0 0 and n is a positive integer. where Eaples: State whether each is a polynoial: 3.) g 5 6.) f ) f ) h 5 5.) f 6 4 Desiré Taylor Math 4 4

5 3. Rational Functions A Rational Function is the quotient of two polynoial functions: A Rational Function is a function of the for f ( ) p( ) an q( ) b n a b n n... a... b a 0 b 0 Asyptotes An asyptote is an iaginary line that the graph of a function approaches as the function approaches a restricted nuber in the doain or as it approaches infinity. I. Locating Vertical Asyptotes p( ) If f ( ) is a rational function, p() and q () have no coon factors and n is a zero of q (), then the q( ) line n is a vertical asyptote of the graph of f (). II. Locating Horizontal Asyptotes n n p( ) an an Let f ( ) q( ) b b... a... b a 0 b 0 i. If n <, then y 0 is the horizontal asyptote ( Botto Heavy ) ii. an If n =, then the line y b iii. is the horizontal asyptote ( Equal Degree ) If n >, there is NO horizontal asyptote. (But there will be a slant/oblique asyptote) ( Top Heavy ) Eaples: Find all vertical and horizontal asyptotes:.) 5 ( ) 3 f.) 3 5 g ( ) 3 3.) 3 7 ( ) 5 h 4.) k ( ) 3 Desiré Taylor Math 4 5

6 4. Trigonoetric Functions sin() cos() tan() csc() sec() cot() 5. Eponential and Logarithic Functions DEFN: An eponential function is a function in the for DEFN: A logarithic function is a function in the for f log y = log b f a. (i.e. the variable is in the eponent) a. (i.e. the variable is in the epression) y is equal to log base b of - Here b is the BASE NUMBER and is the VARIABLE. log b = y eans eactly the sae thing as b y = y = y = log () Coparison of the two graphs, showing the inversion line in red. Desiré Taylor Math 4 6

7 E. Transforations of Functions Vertical Shifts C Horizontal f Shifts f C f C f C Moves Graph UP C units Moves Graph DOWN C units Moves Graph RIGHT C units Moves Graph LEFT C units Vertical and Horizontal Reflections Vertical Stretching/ Copressing c f f Flips Graph About -ais f for c Flips Graph About y-ais Graph Vertically Stretches by a Factor of C c f for 0 c Graph Vertically Shrinks by a Factor of C Eaple: Use the given graph of f() to sketch each of the following. f() a. f() + b. f( + ) c. f() d. f(-) F. Cobinations of Functions. Piecewise-Defined Functions A Piecewise Function is a function that has specific (and different) definitions on specific intervals of. f 0 0 Doain: Range: Desiré Taylor Math 4 7

8 . Sus, Differences, Products and Quotients of Functions Eaple Su f g f g Difference f g f g Product f g f g Quotient. If f() = + and g() = 3 find each of the following. f g f g a. f(4) b. g() c. f(3 4) d. f() + g() e. f()g() 3. Coposition of Functions Eaple: Notation f g f g.) For the functions f and g find a.) f g b.) g f c.) g f d.) g g Desiré Taylor Math 4 8

9 Eaple: For the functions f() and g() given in the graph find a.) c.) e.) g f b.) f g3 f g d.) g f 3 f 4 f f.) g g4 G. Syetry Syetry: Even functions f() = f(- ). Syetric about the y-ais. If (a,b) then (-a,b) Odd functions f(- ) = - f(). Syetric about the origin. If (a,b) then (-a,-b). State whether the following functions are even, odd, or neither. a. f() = b. f() = 4 c. f() = d. f() = sin() e. f() = cos() f. f() = 3 Desiré Taylor Math 4 9

10 H. Function Properties - Increasing functions rise fro left to right Decreasing functions fall fro left to right Positive functions are above the -ais Negative functions are below the -ais *For all of these above, you use the -values to state your answers!. Find each of the following using the given function. a. f() > 0 b. f() 0 c. increasing d. decreasing e. doain and range. Find each of the following using the given function. a. f() > 0 b. f() 0 c. increasing d. decreasing e. doain and range Desiré Taylor Math 4 0