Math 1330 College Algebra Review Part 2 Vertical Line Test A curve is the graph of a function if and only if each vertical line intersects it in at most one point. Example 1: Determine whether the given graph is the graph of a function. A. B. Even and Odd Functions A function f is even if 𝑓𝑓(−𝑥𝑥) = 𝑓𝑓(𝑥𝑥) for all 𝑥𝑥 in the domain of 𝑓𝑓. Since an even function is symmetric with respect to the y-axis, the points (𝑥𝑥, 𝑦𝑦) and(−𝑥𝑥, 𝑦𝑦) are on the same graph. An even function looks the same when reflected about the y-axis. This is the graph of the even function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 . Notice that (−1, 1) and (1, 1) are on the graph. 1 College Algebra Review Part 2 A function f is odd if 𝑓𝑓(−𝑥𝑥) = −𝑓𝑓(𝑥𝑥) for all 𝑥𝑥 in the domain of 𝑓𝑓. Since an odd function is symmetric with respect to the origin, the points (𝑥𝑥, 𝑦𝑦) and (−𝑥𝑥, −𝑦𝑦) are on the same graph. An odd function looks the same when reflected about the x-axis and y-axis or when rotated 180 degrees about the origin. This is the graph of the odd function 𝑓𝑓 = 𝑥𝑥 3 . Notice that (−1, −1) and (1, 1) are on the graph. Example 2: Let (−3, −7) be a point on the graph of 𝑔𝑔. a. If 𝑔𝑔 is an even function, which of the following points is also on the graph of 𝑔𝑔? A. (3, 7) B. (−7, −3) C. (−3, 7) D. (7, 3) E. (3, −7) A. (3, 7) B. (−7, −3) C. (−3, 7) D. (7, 3) E. (3, −7) b. If 𝑔𝑔 is an odd function, which of the following points is also on the graph of 𝑔𝑔? Example 3: Determine if the following function is even, odd or neither. 𝑓𝑓(𝑥𝑥) = 5𝑥𝑥 4 − 3𝑥𝑥 2 Recall: Even: 𝑓𝑓(−𝑥𝑥) = 𝑓𝑓(𝑥𝑥) Odd: 𝑓𝑓(−𝑥𝑥) = −𝑓𝑓(𝑥𝑥) Try this one: Is f ( x) = x 2 + 2 x + 1 even, odd or neither? Recall: Even: 𝑓𝑓(−𝑥𝑥) = 𝑓𝑓(𝑥𝑥) Odd: 𝑓𝑓(−𝑥𝑥) = −𝑓𝑓(𝑥𝑥) 2 College Algebra Review Part 2 A function is increasing on an interval whenever 𝑎𝑎 > 𝑏𝑏 then 𝑓𝑓(𝑎𝑎) > f(𝑏𝑏) (going uphill from left to right). A function is decreasing on an interval whenever 𝑎𝑎 > 𝑏𝑏 then 𝑓𝑓(𝑎𝑎) < f(𝑏𝑏) (going downhill from left to right). The maximum is the largest y value for a function. The minimum is the smallest y value for a function. Example 4: Given the following graph of a function 𝑔𝑔: For parts a – g, state whether the statement is true or false. a. The domain is [3,6). b. The range is (−2, 7). c. The y-intercept is 4. d. The function is decreasing on (0, 1) ∪ (3, 5). e. 𝑔𝑔(𝑥𝑥) = 0 when 𝑥𝑥 = −2 . f. The maximum of the function is 7. g. The minimum of the function is -3. 3 College Algebra Review Part 2 A function is one-to-one (1-1) if no two elements in the domain have the same image. If a function is 1-1 then it has an inverse. 1 The inverse of a function 𝑓𝑓 is denoted by 𝑓𝑓 −1 . Note that 𝑓𝑓 −1 (𝑥𝑥) ≠ 𝑓𝑓(𝑥𝑥). Given the graph of a function, we can determine if that function has an inverse function by applying the Horizontal Line Test. The Horizontal Line Test A function 𝑓𝑓 has an inverse function, 𝑓𝑓 −1 , if there is no horizontal line that intersects the graph in more than one point. Example 5: Which of the following are one-to-one functions? y 1 x −2 1 −1 2 −1 a. b. {(2, -2), (-1, 3), (1, -2)} c. {(-1, 4), (1, 0)} Domain and Range The inverse function reverses whatever the first function did; therefore, the domain of 𝑓𝑓 is the range of 𝑓𝑓 −1 and the range of 𝑓𝑓 is the domain of 𝑓𝑓 −1 . Example 6: Suppose that 𝑓𝑓 and 𝑔𝑔 are 1-1 functions and that 𝑓𝑓(3) = 7, 𝑓𝑓(7) = 2, 𝑔𝑔(7) = 4, and 𝑔𝑔(3) = 0. Find 𝑓𝑓 −1 (𝑔𝑔−1 (4)). 4 College Algebra Review Part 2 Property of Inverse Functions Let f and g be two functions such that (𝑓𝑓 ∘ 𝑔𝑔)(𝑥𝑥) = 𝑥𝑥 for every 𝑥𝑥 in the domain of 𝑔𝑔 and (𝑔𝑔 ∘ 𝑓𝑓)(𝑥𝑥) = 𝑥𝑥 for every 𝑥𝑥 in the domain of 𝑓𝑓 then 𝒇𝒇 and 𝒈𝒈 are inverses of each other. Try this How to find the inverse of a one-to-one function: 1. 2. 3. 4. 5. Replace 𝑓𝑓(𝑥𝑥) by 𝑦𝑦. Exchange 𝑥𝑥 and 𝑦𝑦. Solve for 𝑦𝑦. Replace y by 𝑓𝑓 −1 Verify (i.e. check (𝑓𝑓 ∘ 𝑓𝑓 −1 )(𝑥𝑥) = 𝑥𝑥 AND (𝑓𝑓 −1 ∘ 𝑓𝑓)(𝑥𝑥) = 𝑥𝑥) Example 7: Suppose f is defined for 𝑥𝑥 ≥ 3, by 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 3)2, find the equation of its inverse function. Example 8: Find an equation for of 𝑓𝑓 −1 (𝑥𝑥) given 𝑓𝑓(𝑥𝑥) = −3𝑥𝑥+1 2𝑥𝑥−5 . 5 College Algebra Review Part 2