1. Algebra 2 Inverse Relations and Functions Lesson 7-7

2. Algebra 2 t (sec)f(t)- Velocityg(t)-Distance 000 13216 264 396144 4128256 5160400 Using the Table Find the Following: (a.) f(3) and explain its meaning. (b.) g(4) and explain its meaning. (c.) At what time is the VELOCITY of the object 160 feet/second? Explain how you obtained your answer. (d.) At what time is the DISTANCE (height) of the object 64 feet? Explain how you obtained your answer.

3. Algebra 2 Inverse Relations In other words…If the ordered pairs of a relation ’R’ are reversed, then the new set of ordered pairs is called the inverse relation of the original relation.

4. Algebra 2 Inverse Relations and Functions Lesson 7-7 a.Find the inverse of relation m. Relation m x–1 0 1 2 y–2–1–1–2 Interchange the x and y columns. Inverse of Relation m x–2–1 –1–2 y–1 0 1 2 Additional Examples

5. Algebra 2 Inverse Relations and Functions Lesson 7-7 (continued) b. Graph m and its inverse on the same graph. Relation m Reversing the Ordered Pairs Inverse of m Additional Examples

7. Algebra 2 Function - A function is like a machine: it has an input value that results in a single output. A function is often denoted f (x). No two “x” values can be the same! Vertical Line Test – If for every vertical line on a graph you draw: It goes through only 1 point, y is a function of x. It goes through 2 points (or more), y is not function. Inverse Relations and Functions Lesson 7-7

8. Algebra 2 In mathematics, an inverse function is a function that undoes another function: A function ƒ that has an inverse is called invertible; and it denoted by ƒ −1 : (read f inverse, not to be confused w/exponentiation). Inverse Relations and Functions Lesson 7-7

9. Algebra 2 Inverse Functions Definition: The inverse, f -1 (x), reverses the operations of f (x). If f -1 (x) exists for a certain function f, then f -1 (f (x)) = x. Inverse Relations and Functions Lesson 7-7

10. Algebra 2 One to One (1-1) - A function is called one-to-one if no two values of x produce the same y. No y-values are repeated. So, a function is one-to-one if whenever we plug different values into the function we get different function values. Horizontal Line Test - If every horizontal line you can draw passes through only 1 point, then the function is 1-1. If you can draw a horizontal line that passes through 2 points, then the function is NOT 1-1. Is it Invertible or Not? Inverse Relations and Functions Lesson 7-7

11. Algebra 2 Inverse Relations and Functions Lesson 7-7

12. Algebra 2 Function - A relation in which each input has only one output. Often denoted f (x). Vertical Line Test - If for every vertical line on a graph you draw: It goes through only 1 point, y is a function of x. It goes through 2 points (or more), y is not function. One to One (1-1)- A function is called one-to-one if no two values of x produce the same y. Range The set of y-values Domain The set of x-values Horizontal Line Test - If for every horizontal line on a graph you draw: It passes through only 1 point, then the function is 1-1. It passes through 2 points (or more), then the function is NOT 1-1. FACT: A function has an inverse if and only if it is One-to-One (1-1). To graph an inverse of a function you REFLECT the graph of f over the line y = x Inverse Relations and Functions Lesson 7-7

13. Algebra 2 Change in Domain and Range! Inverse Relations and Functions Lesson 7-7

14. Algebra 2 xf(x)g(x) -262 144 4 6 7-58 10-139 13-2810 Use the Table to answer the following: Inverse Relations and Functions Lesson 7-7

15. Algebra 2 xf(x)g(x) -262 144 4 6 7-58 10-139 13-2810 Use the Table to answer the following: Inverse Relations and Functions Lesson 7-7

16. Algebra 2 Inverse Rule Find an invertible functions inverse: SWAP the variables “x” and “y” SOLVE for “new y”. Using the CHECK Step: −1 (())= to check your work. I like to use x=1 or 0 because the math is simpler Inverse Relations and Functions Lesson 7-7

17. Algebra 2 Inverse Relations and Functions Lesson 7-7

18. Algebra 2 Inverse Relations and Functions Lesson 7-7

19. Algebra 2 Inverse Relations and Functions Lesson 7-7

20. Algebra 2

21. Inverse Relations and Functions Lesson 7-7

22. Algebra 2 Inverse Relations and Functions Lesson 7-7 Find the inverse of y = x 2 – 2. y = x 2 – 2 x = y 2 – 2Interchange x and y. x + 2 = y 2 Solve for y. ± x + 2 = yFind the square root of each side. Additional Examples

23. Algebra 2 The graph of y = –x 2 – 2 is a parabola that opens downward with vertex (0, –2). Inverse Relations and Functions Lesson 7-7 Graph y = –x 2 – 2 and its inverse. You can also find points on the graph of the inverse by reversing the coordinates of points on y = –x 2 – 2. The reflection of the parabola in the line x = y is the graph of the inverse. Additional Examples

24. Algebra 2 Consider the function ƒ(x) = 2x + 2. Inverse Relations and Functions Lesson 7-7 a. Find the domain and range of ƒ. Since the radicand cannot be negative, the domain is the set of numbers greater than or equal to –1. Since the principal square root is nonnegative, the range is the set of nonnegative numbers. b. Find ƒ –1 So, ƒ –1 (x) =. x 2 – 2 2 ƒ(x) = 2x + 2 y = 2x + 2 Rewrite the equation using y. x = 2y + 2Interchange x and y. x 2 = 2y + 2Square both sides. y = x 2 – 2 2 Solve for y. Additional Examples

25. Algebra 2 (continued) Inverse Relations and Functions Lesson 7-7 c. Find the domain and range of ƒ –1. The domain of ƒ –1 equals the range of ƒ, which is the set of nonnegative numbers. d. Is ƒ –1 a function? Explain. For each x in the domain of ƒ –1, there is only one value of ƒ –1 (x). So ƒ –1 is a function. Note that the range of ƒ –1 is the same as the domain of ƒ. Since x 2 0, –1. Thus the range of ƒ –1 is the set of numbers greater than or equal to –1. x 2 – 2 2 > – > – Additional Examples

26. Algebra 2 Inverse Relations and Functions Lesson 7-7

27. Algebra 2 Inverse Relations and Functions Lesson 7-7 The function d = 16t 2 models the distance d in feet that an object falls in t seconds. Find the inverse function. Use the inverse to estimate the time it takes an object to fall 50 feet. d = 16t 2 t 2 = d 16 Solve for t. Do not interchange variables. t = d4d4 Quantity of time must be positive. t = 1414 50 1.77 The time the object falls is 1.77 seconds. Additional Examples

28. Algebra 2 and (ƒ ° ƒ –1 )(– 86) = – 86. Inverse Relations and Functions Lesson 7-7 For the function ƒ(x) = x + 5, find (ƒ –1 ° ƒ)(652) and (ƒ ° ƒ –1 )(– 86). 1212 Since ƒ is a linear function, so is ƒ –1. Therefore ƒ –1 is a function. So (ƒ –1 ° ƒ)(652) = 652 Additional Examples

29. Algebra 2 Problems due for tomorrow: Page 404 #29 (Word Problem) #35-43 odd (No need to use check step) Page 405 #47-57 odd (For all quadratics, just mention that off the bat you know its inverse is NOT a function and stop there.)