2.1 Basics of Functions and Their Graphs
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1 .1 Basics of Functions and Their Graphs Section.1 Notes Page 1 Domain: (input) all the x-values that make the equation defined Defined: There is no division by zero or square roots of negative numbers Range: (output) all y-values that a graph uses. EXAMPLE: Find the domain and range of the following graph (assume graph ends at edge of graph and the bottom edge of the graph is the x-axis, and the left edge of the graph is the y-axis) Domain: [0, 4] Range: [0, ] Function Definition: For each input (x) there can only be one output (y). EXAMPLE: For each relation below, determine whether it is a function. Then give the domain and range for each relation. {(1, ), (3, 7), (, 9), (8, 11)} This is a function. Every x goes to only one y. The domain of this is all the x-values. The answer is {1, 3,, 8}. You may leave it this way or order them. The range of this is all the y-values. The answer is {, 7, 9, 11}. {(-3, 4), (5, 6), (7, 4), (-, 3)} This is a function. Even though the x-values -3 and 7 both go to 4, each x value goes to only one y-value. Domain: {-3, 5, 7, -} Range: {4, 6, 3} Notice that 4 is repeated, but you only need to write it once. {(-, 4), (-1, 6), (0, 3), (-, 8)} NOT a function because when x is - it goes to both 4 and 8. There are two different y values for one x. Domain: {-, -1, 0} Notice again that even though - is repeated, it only needs to be written once. Range: {4, 6, 3, 8} {(5, 3), (-, 1), (5, 3), (9, 10)} This is a function. The same point repeats, but still goes to only one y. Domain: {5, -, 9} Range: {3, 1, 10}
2 Section.1 Notes Page Vertical line test. If you pass an imaginary vertical line through the graph and it only intersects the graph once then it is a function. Which graphs below are functions? Function NOT a function. Function EXAMPLE: Is x y 7 a function? We don t have a graph drawn for us or a set of points. We need to see if it is a function algebraically. First think we need to do is solve for y. We will isolate it and then take the square root of both sides. Don t forget that you will get a plus and minus whenever you take the even root of something. 3y 7 x y 7 3 x 7 x y Notice that for each x we will get two different y values because of the. Therefore we know 3 this is not a function. EXAMPLE: Is x y 5 1 a function? Again we will solve for y. When we do we will take the odd root of both sides. There will be no plus and minus here since it was an odd root. y y 5 1 x 5 1 x To get rid of the fifth power, I took the fifth root. For each x we put in we will only get one y-value for each x we put in so it IS a function. So what is the general rule here based on our previous two examples? Any equation that has a y raised to an even power is NOT a function. Any equation that has a y raised to an odd power IS a function.
3 Section.1 Notes Page 3 Function notation: f (x) which means f of x. This does not mean f times x. It means that we have a function called f which contains the variable x. EXAMPLE: Given the function x 5, find the following: a.) f (3) Whatever is inside the parenthesis goes in place of x in the original expression. This is really asking us for the y value when x is 3. f ( 3) (3) 5 f ( 3) 1 b.) f ( x Now we need to replace x in the original equation with x + 3. Then simplify. f ( x ( x 5 f ( x x 6 5 f ( x x 1 This is as far as we can go on this one. c.) f (3) For this one we can replace the f (x) with x 5. We also know f (3). f (3) x 5 1 f (3) x 4 Notice this is not the same as part b, so the f is not distributed to the x and 3. d.) For this one just replace the x with the expression x + h. ( x h) 5 x h 5 This is as far as we can go. EXAMPLE: Let a.) f (5) x 4. Find the following: x 5 4 f (5) We are replacing x with 5. (5) 1 f ( 5) 7
4 b.) Section.1 Notes Page 4 ( x h) 4 We are replacing x with the quantity (x + h). ( x h) x h 4 This is as far as we can go. x h c.) f (5) x 4 1 x 7 We are replacing f(x) with our original function and f(5) we found in part a. 7 x 4 x 1 7 x x 7 Generally if you have two fractions, then combine after common denominators. 7( x 4) (x 7(x Now add the fractions together now that we have common denominators. 7x 8 x 7(x Distribute and simplify. 9x 1 7(x This is our final answer. EXAMPLE: Given 3 x x, find f ( x). f ( x f 3 x) ( x) ( ) Replace x with x and simplify. 3 ( x) x x We have looked at function notation for equations, but now we will see the relationship between the function notation and graphs. This next exercise shows how to read values off a graph. EXAMPLE: Use the graph below to answer the following: a.) Find the domain Since we don t include the endpoints we have (-, ) (x values) b.) Find the range The answer is (-1, 1] (y-values) c.) Indicate the intercepts x-int: (-1, 0) (1, 0) y-int: (0, 1) d.) Indicate any symmetry this graph has. You can fold this in half over the y-axis, so it has y-axis symmetry.
5 Section.1 Notes Page 5 EXAMPLE: Use the graph below to answer the following: a.) Find f (): This is asking you for the y value when x is -. The answer is f () = 1. b.) Find all x such that This is asking you to find all x that give a y value of 3. This happens at the point (5, 3), so x = 5. c.) Is f ( 3) positive or negative? This is asking you if the y value at x = 3 is above or below the x axis. To find this go over to x = 3. We notice the graph is below the x-axis, so answer is neg. d.) What is the domain? This is asking you for all the x values the graph uses. This would be [-4, 6]. (lowest x to highest x) e.) What is the range? This is asking you for all the y values the graph uses. The answer is [-, 3]. (lowest y to highest y). f.) For which values is 0? This is asking you which part of the graph has positive y values. In other words, what part of the graph is above the x-axis, but not on the x-axis. We have two places this occurs. [-4, 0) or (4, 6) Notice the values I gave in the interval notation are x values. We include the -4 because it is not on the x-axis. More on next page
6 EXAMPLE: Use the graph below to answer the following: Section.1 Notes Page 6 a.) Find f (1): This is asking you for the y value when x is -1. The answer is f (1) =. b.) Find all x such that 0 This is asking you to find all x that give a y value of 3. This happens at x =, 3, and 5. c.) Is f positive or negative? This fraction is the same as When you go to this x value the graph is above the x-axis here, so positive. d.) What is the domain? The domain is referring to the x-values the graph uses. Since there is an open circle at 3, this x-value is not included. So the domain is: [ 3, 6). e.) What is the range? The range is referring to the y-values the graph uses. Again since there is an open circle at 3, this y-value is not included. So the range is ( 4, 4]. f.) Indicate the x and y intercepts. y-int: (0, 3) x-int: (-, 0), (3, 0), (5, 0).
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