Precalculus Notes Unit 1 Day 1

Transcription

1 Precalculus Notes Unit Day Rules For Domain: When the domain is not specified, it consists of (all real numbers) for which the corresponding values in the range are also real numbers.. If is in the numerator and raised to a positive integral. E. f () = 3 or f () = 8 4 Domain: All reals,. If is in the denominator, cannot be any value that makes the denominator zero. E. f E: f () = 4 3. If is inside a square root, values of are restricted to ones that will make the radicand 0. E. f () = 3 E: f () = If is in the square root and in the denominator, values of are restricted to the one that will make the radicand > 0. E. f () = E: f() = 5 Range of a Relation set of y values of a relation Function: special type of relation in which each element of the domain is paired with eactly one element of the range. Testing For Functions Algebraically Solve for y in terms of. If each value of corresponds to eactly one value of y, then y is a function of. Vertical Line Test: tests a graph to see if it is a function.

2 Horizontal Line Test: tests a graph to see if the function s inverse is also a function. E: E: E: Function Notation: If the graph is a function we can use f() instead of y Find f if 3 f.. Find f if f Piecewise Functions: A functions that is defined by two (or more) equations over a specified domain. E: f() = if if 0 0 E: h() = 3 ( ) if if 3 3 Find: f(-3) = f(6) = Find: h(0) = h(-6) = h(-3)= f ( h) h f ( ) The difference quotient, h0, plays an important role in understanding the rate at which a function changes. Using the difference quotient, find and simplify f() = 4 + 3

3 Day f() = 0 0 Graph function by hand: y Graph on calculator y = ( + ) ( < 0) + ( ) ( 0) E: f() = E: f() = 0 0 4) ( y y y Using a calc: Find the relative ma/ and or min of f() = Determine the intervals on which the function is increasing, decreasing, or constant. Find the relative minimum and relative maimum.

4 Determine the intervals on which each function is increasing, decreasing, or constant. a. b. c. d. p()= Even and Odd Functions: A graph has symmetry with respect to the y-ais if whenever (, y) is on the graph then so is (-, y). A graph has symmetry with respect to the origin if (, y) is on the graph then so is (-, -y). A graph has symmetry with respect to the -ais if whenever (, y) is on the graph then so is (, -y). Even Function: A function whose graph is symmetric to the y-ais. f(-) = f() E: y = E: y = E: y = 4 E: y = Odd Function: A function whose graph is symmetric to the origin. f(-) = -f() E: y = /5 E: y = E: y = 4 3 E: y = 3 Determine whether each function is even, odd, or neither. a. y = + b. y = 5 + c. y =

5 Day 3 Toolkit Functions:. y = c. y = 3. y = a 4. y = 5. y = 3 6. y = log a 7. y = 8. y= / 9. y = sin 0. y =. y = [] 3. y = cos Describe Domain & Range Remember y = a(b( h ) + k a < h > 0 a > h < 0 b > k > 0 b < k < 0 Take a look at (Describe the transformations) y = y = 3 y = 3 y = ( ) 3 y = 5 y = y = y = ½ 3 6 y = y = sin y = + 5 y = sin ( + ) y = 3 y = sin 4 y = 4 5 y = 3sin( 4) + 5 Describe transformations : y = ( ) + 4 : y = : y = : y = sin(4 ) +

6 Day 4 5 Combination of Functions: Sum (f + g)() = f() + g() Difference (f g) () = f() g() Product (fg)() = f() g() f f ( ) Quotient ( ) () = g g( ) g() 0 Given: f() = 3 g() = + 4 Find: f() + g() = f() g() = f() g() = f ( ) = g( ) The domain of an arithmetic combination of two functions consists of all real numbers that are common to both functions. f() = 3 g() = + 4 Find:. (f + g) () =. (f g) (-3) = What is the domain:? 3. f(4) g(4) = 4. f (-) = g 5. f() = 6. g() = 4 Composition of Functions: The composition of the functions f with g is denoted by and is defined by the equation (f g)( ) = f(g()) f() = 4 g() = 5. (f g) (4) =. (g f) (-) = 3. (g (f(-3) )= 4. (f (g(3) )=

7 The domain of the composite function is the set of all such that. is in the domain of g and. is in the domain of f. The following values must be ecluded from the input of :. If is not in the domain of g, it must not be in the domain of. Any for which is not in the domain of f must not be in the domain of f() = g() = + 3. (f + g)(). (g + f)() 3. f ( ) g( ) 4. f() g() 5. (f g)( ) 6. (g f)( ). Find each of the following for f() = and g() = 3. Find the domain of each. a. (f g) () b. (g f) (). For: f() = + 3 g() = Domain of (f g) = Domain of (g f) = 3. For : f() = 9 g() = 9 Domain of (f g)() = Domain of (g f)() = 4. For: f() = + 3 g() = ½ ( 3) What is (f g) ()? What is (g f)()? What do we notice?

8 When you form a composite function, you compose two functions to form a new function. It is possible to reverse this process. You can decompose a given function and epress it as a composition of two or more functions. Although there is more than one way to do this, there is often a natural selection that comes to mind. Consider h() = (3 4 + ) 5. Epress the given functions h as a composition of two functions f and g so that h() = (f g)( ) a. h() = 3 b. h() = 3 4 c. h() = 3 Inverses: Let f and g be two functions such that for every in the domain of g and for every in the domain of f. The function g is the inverse of the function f, and is denoted by (read f-inverse ). Thus, and The domain of f is equal to the range of and vice versa. Find the inverse of f informally. Verify that and. a. b. c. The graph of an inverse is the reflection of the original function over the line To have an inverse function, a function must be one-to-one, which means no two elements in the domain correspond to the same element in the range of f. You can use the horizontal line test to determine if a function is one-to-one. Algebraically find the inverse of each function. Then graph the function and the inverse. a. b. c. d.

9 Day 6. Minimum and maimum values are often referred to as values. To approimate etreme values for a function. a. Sketch and label a diagram. b. Write a rule(equation) for the quantity to be minimized or maimized in terms of a single variable. c. Determine the domain for the equation. d. With a graphing calculator, graph the equation and use the function on the calculator to approimate the desired minimum or maimum value. e. Re-read the question and be sure to give the answer for the question that was asked.. Epress the area A of a circle as a function of its circumference C, epress C as a function of A. 3. P(,y) is an arbitrary point on the line a. Epress the distance d from the origin to P as a function of the y- coordinate of P. b. Without graphing, find the minimum distance d and the point P associated with the minimum d. c. What are the domain and range of the distance function? 4. A power station and a factory are on the opposite sides of a river 60 m wide. A cable must be run from the power station to the factory. It costs $5 per meter to run the cable in the river and $0 per meter on land. Use a graphing calculator to find the minimum cost. Power Station 00 m Factory