Contents 1.1 Functions.............................................. 2 1.2 Analzing Graphs of Functions.................................. 5 1.3 Shifting and Reflecting Graphs.................................. 9 1.4 Combining Functions........................................ 13 1.5 Inverse Functions.......................................... 15 1
1.1 Functions Definition 1.1. A function is a rule that establishes a correspondence between two sets of elements (called the domain and range) so that for ever element in the domain there corresponds EXACTLY ONE element in the range. Definition 1.2. A function in one variable is a set of ordered pairs with the propert that no two ordered pairs have the same first element. For eample: { (-2, 1), (-1, 2), (0, 3), (1, 4), (2, 5) }. Definition 1.3. Domain: The things ou can put into a function. Range: The things ou get out of a function. Graphicall An equation defines a function if each vertical line drawn passes through the graph at most once. This is called the Vertical Line Test. For eample: 2
Function Notation We can write a function several was. The variable used to represent elements of the Domain is the independent variable and the variable used to represent elements of the Range is called the dependent variable. The most common wa of writing a function is }{{} = dependent variable f( }{{} ) = 2 }{{ + 1 }. independent variable We can also write a function as or f : 2 + 1 f : {(, ) = 2 + 1} Eample 1.1.1. Which of the following are functions of and wh? 1. 2 + = 1 2. + 2 = 1 3. + 3 = 1 Eample 1.1.2. Find the domain and range of the function f() = + 8. Domain: The values we can put into the function. Range: The values we get out of the function. Eample 1.1.3. Consider the following function: f() = 7+3. Find the values of f(0), f( 1), f( 1+h) and f( + h). 3
Eample 1.1.4. Consider the following function: f() = 4 2 + 3. Find the values of f(0), f( 1), and f( + h). Eample 1.1.5. Suppose f() = 2 2 + 1. Find f( + h) f() h (Difference Quotient) Piecewise Functions Eample 1.1.6. Evaluate f(0), f( 1), f(1), and f(2) for { 2 + 2 if < 1 f() = 2 2 + 2 if 1 4
1.2 Analzing Graphs of Functions Definition 1.4. The graph of a function f is a collection of ordered pairs (, f()) such that is in the domain of f(). Recall: is the distance in the -direction. = f() is the the distance in the direction. Domain and Range The domain of a function is those -values that we can use in the function. The range of a function is the -values we get out of the function. Eample 1.2.1. = 2 Domain: All real numbers. Range: 0. Zero s of a Function Definition 1.5. The zero s of a function f() are those -values for which f() = 0. Q: How do we find the zero s of a function? A: Set the function equal to zero. Also Factor! Factor! Factor! Eample 1.2.2. Find the zero s of f() = 3 2 + 22 16 5
Eample 1.2.3. Find the zero s of f() = 2 9 + 14. 4 Increasing and Decreasing Functions Definition 1.6. A function is increasing on an interval if for an 1 and 2 in the interval with 1 f( 1 ) < f( 2 ). A function is decreasing on an interval if for an 1 and 2 in the interval with 1 f( 1 ) > f( 2 ). A function is constant on an interval if for an 1 and 2 in the interval f( 1 ) = f( 2 ). < 2 then < 2 then Eample 1.2.4. (-1,2) (1,-2) 6
Linear Functions Graph: f() = m + b Linear Function Eample 1.2.5. Graph f() = 3 2 2 Step 1: Plot - intercept. Step 2: Plot another point using the slope 3 2 = rise run Graphing Piecewise Functions Eample 1.2.6. Graph f() = 3 2 2 2 3 2 + 7 < 2 7
Eample 1.2.7. Graph 0 f() = 0 0 < 1 1 > 1 Even and odd functions Definition 1.7. A function is even if f() = f( ) for all in the domain of f(). A function isodd if f() = f( ) for all in the domain of f(). Eample 1.2.8. Is h() = 5 5 3 even, odd or neither? Look at h( ): h( ) = ( ) 5 5( ) 3 Eample 1.2.9. Is h() = 2 even, odd or neither? Look at h( ): h( ) = ( ) 2 Eample 1.2.10. Is h() = 3 5 even, odd or neither? Look at h( ): 8
1.3 Shifting and Reflecting Graphs Basic Graphs f() = f() = 2 f() = f() = 3 Shifting Graphs Moving up and down h() = f() + a h() = f() a moves f() up a units. moves f() down a units. Eample 1.3.1. h() = 3 3 = f() 3 if f() = 3. Graph f() and h() on the same set of aes. 6 3 3 2 1 3 6 9 f() = 3 1 2 3 4 h() = 3 3 9
Moving left and right h() = f( + a) h() = f( a) moves f() to the left a units. moves f() to the right a units. Eample 1.3.2. h() = 2 = f( 2) if f() =. Graph f() and h() on the same set of aes. 4 3 2 1 2 1 2 3 f() = 2 4 h() = 2 Eample 1.3.3. h() = ( + 2) 3 3 = f( + 2) 3 if f() = 3. Graph f() and h() on the same set of aes. 8 f() = 3 6 4 2 3 2 1 1 2 3 4 2 4 6 8 10
Reflecting across -ais h() = f() reflects f() across the -ais. Eample 1.3.4. h() = 2 = f() if f() = 2. Graph f() and h() on the same set of aes. 4 f() = 2 3 2 1 4 2 1 2 4 2 3 4 Eample 1.3.5. h() = ( + 2) 2 + 2 = f( + 2) + 2 if f() = 2. Graph f() and h() on the same set of aes. 4 f() = 2 3 2 1 6 4 2 1 2 4 2 3 4 5 6 11
Eample 1.3.6. Write the equations of the following graphs. 6 5 4 3 2 1 2 1 2 3 4 5 6 2 4 6 Eample 1.3.7. Write the equation for the function that has the shape of f() = 2 but is shifted 3 units to the left, 7 units up and then reflected across the ais. 12
1.4 Combining Functions Arithmetic Combinations We can add, subtract, multipl and divide functions much like we do with real numbers. Notation 1. (f + g)() = f() + g() 2. (f g)() = f() g() 3. (f g)() = f() g() ( ) f 4. () = f() g g() Eample 1.4.1. If f() = 2 + 3 and g() = 2 + 1 find (f + g)() = 2 + 3 + 2 + 1 = 2 + 2 + 4 (f g)() = (f g)() = ( ) f () = g We can evaluate these new functions the eact same wa we did before. Whatever is in the parentheses is replaced for in the equation. Eample 1.4.2. If f() = 2 + 2 3 and g() = 3 3 2 4 find a) (f + g)( 1) = b) (f g)(2) = c) The domain of ( ) f () = g 13
Composition of Function The composition of a function f with a function g is (f g)() = f(g()). The domain of (f g) is the set of all in the domain of g such that g() is in the domain of f. Eample 1.4.3. Suppose f() = 3 + 2 + 1 and g() = 1 then find a) (f g)() = f(g()) = f( 1) b) (g f)() = g(f()) = g( 3 + 2 + 1) c) (f f)() = f(f()) = f( 3 + 2 + 1) Eample 1.4.4. Find (a) (f g)(), (b) (g f)() and (c) the domain of each for f() = 4 and g() = 2. Eample 1.4.5. Find functions f and g such that h() = 3 2 4 = (f g)(). 14
1.5 Inverse Functions One to one functions A function is said to be one to one (1-1) if no two ordered pairs have the same second component but different first component. A function has one value for each value but those values can repeat. In a 1-1 function the values never repeat. Graphicall: An equation must pass the Vertical Line Test to be a function. A function must pass the Horizontal Line Test to be 1-1. Eample 1.5.1. f() = 2 is not 1-1. f() = 3 is 1-1. Inverses The indentit function is f() = or =. You get out what ou put in. Given a function f that is 1-1 then f has an inverse f 1 and If f is not 1-1 then f 1 DOES NOT EXIST. f(f 1 ()) = and f 1 (f()) =. 15
Graphicall = 2 = Eample 1.5.2. Draw the inverse of the following function on the same set of aes. 16
Finding Inverses Algebraicall Eample 1.5.3. Find the inverse of f() = 2 + 4 on 2 Step 1: Solve for. = 2 + 4 4 = 2 4 + 4 4 = ( 2) 2 Step 2: Check the domain. Step 3: Switch and. 2 so use = 2 4 Step 4: Write f 1 () = 2 4. Step 5: Check that f(f 1 ()) =. Eample 1.5.4. Find the inverse of f() = 4 2. 17