1. Algebra-2 Weak Area Review Part 2

2. Your turn: 1. Which of the 3 functions restrict domain? 2. Which of the 3 functions restrict range?

3. What can’t the radicand equal ? (for even index numbers) The even root of a negative number does The even root of a negative number does not have a real solution. not have a real solution. The radicand (of an even root) cannot be a negative number in order to have a real solution.

4. Your turn: 3. What is the domain? 4. What is the range?

5. Function Notation y = f(x) “y is a function of x” ‘y’ equals ‘f’ of ‘x’ A function is a rule that matches input values to out put values. f(x) = 2x + 1 (Input) x(rule) 2x + 1 2x + 1(output) y 2 2(2) + 1 5 f(2) = 5 3 2(3) + 1 7 f(3) = 7

6. Functions f(x) = 2x  f(3) = ? Means: wherever you see an ‘x’ in the function, replace it with a 3. f(3) = 2(3) 1. Replace the ‘x’ with a set of parentheses. f(3) = 2( ) 2. Put the input value ‘3’ into the parentheses. 3. Find the output value. f(3) = 6

7. Functions  f(2) = ? Means: wherever you see an ‘x’ in the function, replace it with a ‘2’. 1. Replace the ‘x’ with a set of parentheses. 2. Put the input value ‘-2’ into the parentheses. 3. Find the output value. f(2) = 0 Cool, we found a zero of the function. zero of the function.

8. Your turn: 5. 6. 7.

9. Composition of Functions If your input is a function instead of a number input the expression into the instead of a number input the expression into the function and just apply the rule to it. function and just apply the rule to it. f(x) = 2x + 1 (Input) (rule): f(x) 2x + 1 2x + 1 (output) f(1) = 3 g(x) 2(g(x)) + 1 1 – 3x 1 – 3x f(g(x))= f(1 – 3x) = 3 – 6x g(x) = 1 – 3x 1 2(1) + 1 3 2(1 – 3x) + 1 3 – 6x

10. Compositions of Functions f(x) = 2x + 3 and f(g(x)) = ? 1.Replace the ‘x’ in f(x) with a set of parentheses. a set of parentheses. 2.Put the input value (g(x)) into the parentheses. into the parentheses. 1.The input value to f(x) is g(x). 3. Find the output value.

11. Compositions of Functions f(x) = 2x - 3 and 1.Replace the ‘x’ in g(x) with a set of parentheses. a set of parentheses. 3. Put the input value (f(x)) into the parentheses. into the parentheses. 2. The input value is f(x). 4. Find the output value. Another way to write a composition.

12. Your turn: 8. 9. 10.

13. Function compositions One more layer. The input to g(x) is 2. The input to f(x) is g(2), so the input to f(x) is 4. input to f(x) is 4.

14. Your turn: 12. 11. 13.

15. Vocabulary Inverse Relation: A relation that interchanges the input and output values of the original relation. the input and output values of the original relation. (-2, 5), (5, 6), (-2, 6), (7, 6) Relation: Inverse Relation: (5, -2), (6, 5), (6, -2), (6, 7)

16. Graphs of Inverse Relations y = x Each point in the inverse relation is a point from the relation is a point from the relation reflected across the line y = x reflected across the line y = x

17. How to find the inverse relation Relation: 1. Exchange ‘x’ and ‘y’ in the original relation. 2. Solve for ‘y’ (get ‘y’ all by itself).

18. Your Turn: Find the inverse of: This is the inverse of: y = 4x + 2 14. y = 4x + 2 Exchange ‘x’ and ‘y’ subtract ‘2’ (left and right) Divide (all of the) left and right by 4 Reduce the fractions Rearrange into “slope intercept form”

19. Your Turn: Find the inverse of: 15. Exchange ‘x’ and ‘y’ Add ‘3’ (left and right) Square root both sides This is the inverse of: Simplify Which is the same thing as:

20. How to write: “the inverse of f(x)” means “the inverse function of f(x) means “the inverse function of f(x) Do not confuse this notation with the negative inverse property: The inverse of a number means “flip the number (the reciprocal of the number)” The inverse of a function means “exchange ‘x’ and ‘y’ then solve for ‘y’.”

21. Your Turn: 16. Are f(x) and g(x) inverses of each other ? 17. Are f(x) and g(x) inverses of each other ?

22. Your Turn: 18. Solve 19. Solve

23. Your Turn: 20. Find the inverse Log = =

24. Your Turn: 21.

25. 21.

26. How can you tell if the inverse of a function is a function? Horizontal Line Test: if the line intersects the graph more than once, then the Inverse of the function is NOT a function.

27. Your turn: Calculate ‘x’. 23.

28. Your turn: Expand Condense 24. 25.

29. 26. The front row of a rock concert has a sound intensity of of The reference sound intensity The reference sound intensity What is the sound level in decibels on the front row of the rock concert?

30. Your turn: A solution has hydrogen-ion concentration of moles/liter. What is the pH of the moles/liter. What is the pH of thesolution? pH = -log [H+] pH = -log [H+] 27.

31. Your turn: Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. 28. Some storm chasers measured the speed of the wind inside a tornado. It was 275 mph. How far will the tornado travel along the ground?. (1) Plug numbers into the formula (2) Solve for the unknown variable in the formula

32. Your turn: A bank account earning 4% interest has $3569 in it. The original deposit was $2000. How long has the money been in the account? 29. Log = =

33. Using an inverse function to solve an equation. Ticket prices in the NFL can be modeled by: where ‘t’ is the number of years since 1995. years since 1995. During what year was the price of a ticket $50.85 ? (price as a function of time since 1995)

34. Your Turn: 30. Solve: 31. solve:

35. Your Turn: 7 A C B 32. 33. 3 5.7 F D E 3.2

36. Your turn: describe the transformations of f(x) 33. Period = ? 34. Horizontal translation (phase shift) = ? 35. Vertical translation = ? 36. Amplitude = ? 37 frequency = ?