Regents Review Session #3 Functions A relation is a set of ordered pairs. A function is a relation in which each element of the domain corresponds t exactly one element in the range. (Each x value is paired with one and on one y value.) I. What do you do when you have to identify a function? If given ordered pairs make sure every x goes to only one y. If given a mapping make sure every element in the domain 'circle' g to only one element in the range circle If given an equation you must be able to identify the equation and t determine if the equation or the graph of the equation is a function. If you are given a graph of an equation use the vertical line test. If a vertical line can be drawn anywhere within the domain of the function an intersect your graph only once, you have a function. Examples: Function or not? 1. {( 7, 2), (4, 2), ( 3, 9), (5, 1)} 2. {(4, 7), (6, 2), ( 3, 7), (9, 2)} 2b {(4, 7), (6, 2), ( 3, 7), (4, 2)} 3. y = x 2 4. x = y 2 5. y = 5 6. x = 5 7. x 2 + y 2 = 25 8. 2 9. 2 3 4 4 3 4 4 5 5 10. y = 2 x 3 2 11. y = 3sin2x 12. y = 2 x 3 13.y = log 2 x 14. y = (x + 4) 1/2 1
Domain all possible values of x that can be used to ensure that the function makes sense Range all possible y values that you can obtain from the function CONCERNS FOR DOMAIN: 1. You cannot have a negative under the radical (We are using the real number system when we discuss domain) 2. You cannot have a zero in the denominator of a fraction (undefined value) 3. You cannot have a log answer position of zero or a negative number. (Remember log base answer = exponent) 2
II. What do you do when you have to identify the domain of a function? A. If you have a fraction with a variable in the denominator 1. set the denominator equal to zero. 2. Solve the equation 3. The solution you get must be restricted from the domain. All real numbers except those solutions are valid for the domain. B. If you have a square root 1. Set what is inside the radical greater than or equal to zero. 2. Solve the inequality 3. The solution to the inequality is the domain for the function. C. If you have a fraction with a variable under a radical in the denominator of a fraction 1. Set what is inside the radical greater than zero 2. Solve the inequality 3. The solution to the inequality is the domain for the function. D. If you have a log expression/equation where there is a variable in the answer position 1. Set the answer position greater than zero. 2. Solve the inequality 3. The solution to the inequality is the domain for the function E. If you have a set of ordered pairs 1. The x values are the domain, the y values are the set of range. 3
Examples: Find the domain. 1. y = x + 7 2. y = 12 x 2 + 2x 24 3. log 4 (2x + 19) 4. y = x 2 + 7 5. y = 2x 8 6. y = 5x x 2 4 7. y = 2 x 2 + 3x + 2 4
Given each of the graphs, state the domain and range of each, using interval notation. y y 4 x 2 2 4 x 5
Basic Graphs for Linear, Quadratic, Absolute Value, Radical and Exponential Linear y = mx + b Change the slope change the steepness of the line Change the y intercept change the place where the line cross the y axis. Function or Not? Sketch each (roughly). Domain and Range, in interval notation. 1. y = 3x 4 2. y = 3x + 8 y = 3x 4. y = 5 5. x = 2 6
Quadratic Two forms: standard form: y = ax 2 + bx + c Vertex form: y = a(x h) 2 + k, where (h, k) is the vertex Function or Not? Sketch each (roughly). Domain and Range, in interval notation. 1. y = x 2 + 10x 3 2. y = 2x 2 14x + 1 3. y = x 2 4. y = (x 4) 2 5. y = (x + 3) 2 6. y = ½ (x - 1) 2 7. y = 3(x - 1) 2 8. y = -2(x - 1) 2 9. y = x 2 10. y = x 2 + 5 11. y = x 2 4 12. y = (x 3) 2 + 7 13. y = (x + 2) 2 5 7
Absolute Value: y = a x h + k, where (h,k) is the turning point Function or Not? Sketch each (roughly). Domain and Range, in interval notation. 1. y = x 2. y = ½ x 3. y = 2 x 4. y = x 4 5. y = x + 2 6. y = x + 9 7. y = x 5 8. y = 3 x + 1 6 9. y = x +7 + 2 8
Radical: where (h, k) is your 'starting' point Function or Not? Sketch each (roughly). Domain and Range, in interval notation. 1. 2. 3. 4. 5. 6. 9
Exponential: y = ab x Function or Not? Sketch each (roughly). Domain and Range, in interval notation. 1. y = 2 x 2. y = 2 x + 3 3. y = 2 x 4 4. y = 2 x+ 1 5. y = 2 x 3 10
III. What do you do if you have to evaluate a function? Remember, function notation looks like this f(x) this does not mean t multiply f and x, it is read as f of x and it means the value of the function when x is the input. Therefore, f(2) means the value of the function when x = 2. 1. Plug the given value of x into the right side of the function and evaluate. 2. If the value you are plugging in is an expression and not a number, b careful to distribute/multiply properly. 3. If you are asked to find x when f(x) equation something, be sure to substitute properly. 1. If the function f(x) is defined as f(x) = x 2 + 2x 4, evaluate f(3). 1. Using the graph shown, find: f(2), f( 3), and f(9 ). 2. If f(x) is represented by the following mapping, find: f(3), f(0), an f( 2) 3 12 2 10 1 7 0 5 1 2 2 5 3 1 3. Given f(x) = x 2 3x 1, find x such that f(x) = 3. 4. If the function g(x) = 2 + 8x 2x 2, evaluate g(5). 5. Explain, in terms of x and y, how the statement f( 1) = 5 translates into words. 11
IV. Operation with Functions What do you have to do if you have to add, subtract, multiply or divide fractions? Add/Subtract 1. You are just combining like terms, as you would have with any other algebraic expression. Multiply 2. Distribute/foil and then combine like terms. Dividing 3. Simplify the fraction if possible. Example. 1. Let f(x) = 2x 2 + x 3 and g(x) = x 1 Perform each operation and then find the domain. a. f(x) + g(x) b. g(x) f(x) c. f(x) g(x) d. f(x) * g(x) e. f(x) g(x) 12
V. What do you do when you have to do a composition of function? 1. You always start at the end and work your way backwards or start inside and work your way out. 2. Evaluate the first function then use that output as the input for the second function. 3. Remember f o g (x) is read as f following g of x Examples. 1. Let f(x) = x 2 + 4, g(x) = 2x, and h(x) = x 3, find: a. h(f(2)) b. (g h)( 5) c. g(f(0)) d. (f g)(x) e. g(h(x)) f. (f h)(x) 2. Is the composition of functions commutative? Give an example to support your answer. 13
One to One A function is one to one if every input has exactly one output and every output comes from exactly one input. A one to one function must pass the vertical AND horizontal line tests. Sometimes the domain is restricted to limit the domain of the graph. Examples: One to one? 1. y = x 2 2. y = 3x 5 3. y = x 3 4. y = x 5, if x > 5 5. y = cosx 6. y = cos x if 0 < x < 180 A function from A to B is called onto for all b in B there is an a in A such that f (a) = b. All elements in B are used. (If you can use it as an input, you must be able to get it out for an output). The sets may also be restricted. Examples: Onto? 1. f(x) = 4x 1 as function f(x) maps from Reals to Reals 2. g(x)= x 2 + 3 3. h(x) = x + 4 as function h(x) maps from the Reals to Reals 4. h(x) = x + 4 as function h(x) maps from the Reals to [0, ) 14
VI. What do you do when you have to find the inverse of a function? 1. Take the function notation out and replace it with y. 2. Switch x and y. 3. Solve for x 4. Rewrite the answer in function notation f 1 (x). Note: If you have to evaluate using the inverse, find the inver and then follow the steps discussed on how to evaluate a function. Examples: 1. Find the inverse of f(x) = 2x + 4 3 2. Find f 1 (x) if f(x) = x 3 4 3 3. Find g 1 (x) given g(x) = {(3, 4), (2, 9), ( 5, 7)} 4. If f(x) = 3x + 1, state f 1 (x) and evaluate f 1 (4)? 6. If h(x) = ½ x 3 and g(x) = 2x 1, evaluate g(h 1 ( 3)) 15
VII. What do you do if you have to prove a two functions, f(x) and g(x) are inverses of each other? 1. Do the composition f(g(x)). 2. Do the composition g(f(x)). 3. If the answers to step 1 and step 2 each equal x, the two functions are inverses of each other. Example: Prove f(x) and g(x) are inverse functions. f(x) = 2x 9 g(x) = x + 9 2 16
Equation of a Circle (x h) 2 + (y k) 2 = r 2 where (h,k) is the center and r is the length of the radius Identify the center and the radius in simplest radical form. 1. x 2 + (y 2) 2 = 20 2. (x + 4) 2 (y 9) 2 = 45 Write in center radius form and identify the center and the radius. 1. x 2 + y 2 4x + 8y 2 = 0 2. 2x 2 + 2y 2 + 16x 20y 4 = 0 17
Direct Variation and Inverse Variation. Direct Variation: Two variables that are directly proportional increase or decrease by the same factor. When x and y vary directly, If x and y vary directly, when x = 4, y = 12, find y and x = 11. Indirect or Inverse Variation: When two variables are inversely proportional, as one increases, the other decreases but the product of the two variables remain constant. When x and y vary inversely, xy = xy. 1. If x and y vary inversely when x is 14, y is 6, what is x when y = 4? 2. The cost of a limo for the prom is $420 regardless of how many people are in the limo. If 6 people are in the limo, the cost per person is $70. How much would the limo cost each person if there were 10 people? 18
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