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1 page 1 of 16 CLASS NOTES: 3 8 thru 4 3 and 11 7 Functions, Exponents and Polynomials 3 8: Function Notation A function is a correspondence between two sets, the domain (x) and the range (y). An example of a function is f ( x ) = 2 x + 3 f ( x ) is the same as y The domain refers to x. x is called the independent variable because it can be ANY number well mostly! The range refers to y. y or f x ( ) is called the dependent variable because its value depends on the value of x that we plug in. These mean the same thing!! Evaluate the given expression if x = 4. Ex 3 x 1 If f ( x ) = 3 x 1, find the indicated value. ( ) Ex f 4 If x = 4, then 3 x 1 = 3 ( 4 ) 1 = 12 1 = 11 If f ( x ) = 3 x 1, then ( ) = 3 ( 4 ) 1 f 4 = 12 1 = 11 Therefore, f ( 4 ) = 11

2 page 2 of 16 EX 1 Find the indicated values for the function f ( x ) = x (a) f ( 3 ) = Hint: Sometimes you may see functions written as: f : x x (b) f ( 5) = (c) f ( 1 ) = (d) f ( 0 ) = (e) f ( a ) = (f) f ( 3b ) = Finding the RANGE of a function EX 2 Given f ( x ) = x 3 with the domain D = { 0, 1, 2, 3, 4}, find the range of f ( x ). Plug in each of the five values for x. Those five answers are the members of the range, R = {,,, }. Even if a range value occurs twice, you only include it once in the range set.

3 page 3 of 16 Finding the DOMAIN of a function This means finding the values that x can be. This may seem strange since x is the value that we plug into the function. You may be thinking, Don t they usually tell us what to plug in? And yes, they do! However, finding the domain means asking ourselves, Are there any numbers that I cannot plug into the function? And yes, there might be! There are TWO situations that may cause us to exclude values from the domain. [ 1] If the function is a fraction, any value of x that makes the denominator = zero must be excluded from the domain. Ex If f ( x ) = 2 x, then x 3 because the denominator would be 0. x 3 So, the domain of this function is R except x 3 [ 2 ] If the function includes a square root, then the domain is the set of values that make the value of what is under the square root sign 0. Ex If f ( x ) = 2 x + 1, then 2 x x 1 x 1 2 So, the domain of this function is x 1 2 If neither of these two situations apply, then the domain is simply R (all real numbers). EX 3 Give the domain of each function. (a) f ( x ) = x 2 x + 1 (b) f ( x ) = 3 4 x + 5 6

4 page 4 of 16 EX 4 Give the domain of each function. (a) f ( x ) = x 7 3 (b) f ( x ) = 2 x x (c) f ( x ) = x ( x + 2 ) x 3 ( ) (d) f ( x ) = 3 x 4 (e) f ( x ) = x + 5 (f) f ( x ) = x x

5 page 5 of 16 COMPOSITION OF FUNCTIONS This means TWO or more functions combined by +, --, or a function inside of a function EX 5 Given f ( x ) = 3 x 2 and g x (a) f ( 1 ) + g ( 2 ) = ( ) = x 2, find the indicated values. (b) f ( g ( 5 )) = Hint: Find the value of the innermost function first!! (c) g ( f ( 1 )) = (d) f ( f ( 3 )) = (e) g ( 2 f ( 1) ) = (f) f ( g ( a )) =

6 page 6 of : Linear Functions LINEAR FUNCTIONS Functions that are in the form f ( x ) = mx + b. m is the slope of the line. ( 0, b ) are the coordinates for the y-intercept. This is also denoted as f ( 0 ) = b f ( x ) = y Function Notation: OLD Equation Notation y = 2 x 1 f x NEW Function Notation ( ) = 2 x 1 Complete the ordered pair ( 4, ) to form a solution for the equation. Plug in 4 for x and solve for y. Find the value for f ( 4 ). Plug in 4 for x and simplify. f ( 4 ) = 2 ( 4 ) 1 = 8 1 = 7 The solution is ( 4, 7) f ( 4 ) = 7 EX 1 Find an equation of each linear function f using the given information. (a) m = 4, b = 3

7 page 7 of 16 EX 2 Find an equation of each linear function f using the given information. (a) m = 1 2, f ( 0 ) = 5 (b) slope is 1, f ( 2 ) = 3 Plug in what we know Because f ( x ) = y, ( ) = 3 is the same as the point 2, 3 f 2 x y ( ) f ( x ) = mx + b ( ) = 1 ( 2 ) + b f 2 3 = 2 + b 5 = b We need m and b [aka f 0 f ( x ) = mx + b Answer: f ( x ) = 1 x + 5 ( ) ] to write a linear function in the form (c) m = 4, f ( 2 ) = 7 (d) f ( 3 ) = 12, f ( 5 ) = 2 Hint: m is y 2 y 1 x 2 x 1 = f ( 5 ) f ( 3 ) = 5 3 = 14 = 7 2

8 page 8 of 16 EX 3 Find an equation of each linear function f using the given information. (a) f ( 3 ) = 2, f ( 12 ) = 4

9 page 9 of : Polynomials Examples: 3 x 4 + x 3 y + 4 x 2 y 2 2 xy 3 + 9y x 3 6 x 2 + x 4 Constant - a number (without a variable) Monomial - a constant, a variable or a product of constants and variables. Term another word for monomial Binomial the sum of two monomials. Polynomial - the sum of two or more monomials. Coefficient - the number (factor) in front of the variables of a monomial. Leading Coefficient the coefficient of the term with the highest power. Degree of a Variable the power (exponent) of a variable in a monomial. Degree of a Monomial the sum of the powers of the variables in a monomial. Degree of a Polynomial the largest of the degrees of the monomials in a polynomial. Descending Order If the polynomial is a single variable (one variable) polynomial, the terms are arranged so that the exponents of the variable decrease from left to right. Similar Monomials (also called like terms ) monomials that have the exact variables and powers of those variables. Combine Similar Terms by adding their coefficients Simplified Polynomial to simplify a polynomial (1) combine similar terms and (2) arrange terms in descending order.

10 page 10 of 16 Given the example: 7 x x + 4 x 3 1. Draw a circle around each monomial. 2. Rewrite the polynomial in descending order. 3. What is the value of the constant? 4. What is the coefficient of x? 5. What is the leading coefficient? 6. What is the degree of the term 7 x 2? 7. What is the degree of the polynomial? 8. What is the degree of the term 5 x 2 y 2? 9. What is the degree of the term 9 xy 2? 10. Which term is similar to 7 x 3? 11. Simplify the polynomial x 3 x x x ADDING and SUBTRACTING polynomials is as simple as combining similar terms. EX: Add 2 x 2 3 x + 5 and x 3 5 x x 5 EX: Subtract 2 x 2 3 x + 5 from x 3 5 x x 5

11 page 11 of : Using Laws of Exponents Simplify. EX: x 5 x 3 x a x b = x a +b EX: EX: ( xy ) 3 ( 5 x ) 2 ( xy ) a = x a y a EX: $ x 2 % ' 4 $ x a % ' b = x a b EX: $ x 3 y 5 % ' 4 EX: $ 2 x 2 y 3 % ' 5 $ x a y b % ' n = x an y bn EX: % x 3 $ ' ( 2 EX: $ xy 2 % ' 3

12 page 12 of 16 Simplify completely. EX: x $ x 2 % ' 3 x 5 EX: $ 3 xy 2 z 3 % ' 3 EX: % 3 x 2 y 3 $ ' ( % 4 xy 2 $ ( EX: 3 x ' ( ) $ 4 x 5 % ' EX: % x 2 $ ' ( % x 4 $ ( EX: x 2 x k ' EX: $ x 2 % ' k EX: $ x 2 % ' k $ x k % ' 3

13 page 13 of : Multiplying Polynomials (Using FOIL) FIRST OUTER INNER LAST Then COMBINE like terms! Multiply ( x + 2 )( x + 5) F + O + I + L x x x 5 2 x 2 5 x 2 5 x 2 x 10 x x + 10 EX 1 Multiply. ( x + 2 ) ( y ) EX 2 Multiply. ( x + 2 )( 3 x + 1) EX 3 Multiply. ( 2a 5) ( 3a + 4) EX 4 Multiply. 3 x + 1 ( ) $ x x + 6 % '

14 page 14 of 16 ( a + b ) 2 does NOT equal a 2 + b 2!!!!!!! Use F-O-I-L to multiply the following. ( a + b ) 2 ( a b ) 2 ( a + b )( a b ) EX 5 Multiply. ( x + 1) 2 EX 6 Multiply. ( x 3) 2 EX 7 Multiply. ( x + 5) ( x 5)

15 page 15 of : Powers of Binomials (Binomial Expansion) ( a + b) 0 = 1 ( a + b) 1 = 1a 1 + 1b 1 ( a + b) 2 = 1a 2 + 2ab + 1b 2 ( a + b) 3 = 1a 3 + 3a 2 b + 3ab 2 + 1b 3 ( a + b) 4 = 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + 1b 4 ( a + b) 5 = 1a 5 + 5a 4 b + 10a 3 b a 2 b 3 + 5ab 4 + 1b 5 Binomial Expansion - The sum of terms that you get when you multiply out a power of a binomial. The pattern of coefficients comes from a triangular array of numbers called Pascal s Triangle. EX 1 Expand ( a + b) 6 by using row 6 of Pascal s Triangle. EX 2 Find the first four terms in the expansion of ( x 2y) 7.

16 page 16 of 16 EX 3 Find the first three terms of the expansion of ( 2x y) 6. EX 4 Expand and simplify the expression ( x 3) 4. EX 5 The first three terms of the expansion of a + b ( ) 20 are a a 19 b + 190a 18 b 2. Write the last three terms.

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