2. Solve. 1. x 5/2 = 32 2. x 2/3 + 15 = 24 3. 4x 3/4 = 108 x 3/4 = 27 (x 5/2 ) 2/5 = 32 2/5 x 2/3 = 9 x = 2 2 x = (32 1/5 ) 2 x = 4 (x 2/3 ) 3/2 = 9 3/2 x = (9 1/2 ) 3 x = 3 3 x = 27 (x 3/4 ) 4/3 = 27 4/3 x = (27 1/3 ) 4 x = 3 4 x = 81

3. 7.5 Solving Radical Equations Objective - To be able to solve square root and other radical equations. State Standard 15.0 - Students will be able to solve radical expressions

4. Solve: x – 4 = 0 3 ( x) = 4 ( x) = 4333 x = 64 x = 64 Example 1: x = 4 3 7.5 Solving Radical Equations

5. Example 2: Solve: 2x 3/2 = 250 x 3/2 = 125 (x 3/2 ) 2/3 = 125 2/3 x = 125 2/3 x = 25 x = (125 1/3 ) 2 x = 5 2

6. Example 3: Solve: 3x + 2 – 2 x = 0 3x + 2 = 2 x ( 3x+2) 2 = ( 2 x ) 2 3x + 2 = 4x 2 = x 2 = x

7. Example 4: Solve: x – 4 = 2x (x – 4) 2 = ( 2x ) 2 x 2 – 8x + 16 = 2x x 2 – 10x + 16 = 0 (x – 2)(x – 8) = 0 x – 2 = 0 and x – 8 = 0 x = 2 and x = 8

9. 7.6 Warm-Up Perform indicated operations 1.(2x+7) 2 2. 2 2 4x 2 +28x+49 4

10. Operations on Functions Operation Definition Example f(x)=2x, g(x)=x+1 Addition f(x)+g(x) 2x+(x+1) = 3x+1 Subtraction f(x) – g(x) 2x – (x+1) = x – 1 Multiplication (f(x))(g(x)) 2x(x+1) = 2x 2 + 2x 7.6 Function Operations f(x) 2x g(x) (x+1) Division

11. Example 1 2x - 2 Adding and Subtracting Functions f(x)=3x, g(x)=x+2 f(x)+g(x)f(x) - g(x) 3x+(x+2) 3x - (x+2) 4x+2

12. Example 2 3x 2 +6x 3x/(x+2) Multiplication and Division Functions f(x)=3x, g(x)=x+2 (f(x))(g(x)) (3x)(x+2) f(x)/g(x)

13. h(x) = -5x + 16 f(x) = x + 4, g(x) = 3x h(x) = 2(f(x)) + 2(g(x))h(x) = 4(f(x)) - 3(g(x)) h(x) = (2x+8) + (6x)h(x) = (4x+16) - (9x) h(x) = 8x + 8

14. h(x)=3x-3 / 4x h(x)=(-2(f(x))(g(x)) f(x)=x-1, g(x)= 2x h(x)=3(f(x)) / 2g(x) h(x)=3(x-1) / 2(2x) -4x 2 + 4x h(x)=(-2x+2)(2x) h(x)=(-2(x-1))(2x)

15. Composition of two functions The composition of the function f with the function g is: f(g(x)) or (f ο g)(x) This is read as: f of g of x

16. Example 3a f(x) = 2x and g(x) = 3x + 1 2(3x + 1) f(3x + 1) f(g(x)) 6x + 2 3(2x) + 1 g(2x) g(f(x)) 6x + 1

17. Example 3b f(x) = 3x -1 and g(x) = 2x – 1 3(2x-1) -1 f(2x-1) f(g(x)) 3 2x-1

18. Example 3c f(x) = x 3 and g(x) = x 2 + 7 8 2 + 7 g(x 3 ) (g ο f)(2) g(f(x)) g(2 3 ) g(8) 64 + 7 71

19. Evaluate the compositions if: f(x) = x + 2g(x) = 3h(x) = x 2 + 3 1. f(g(x))2. h(f(x))3. h(f(g(x))) f(3) 3 + 2 5 h(x + 2) (x + 2) 2 + 3 x 2 + 4x + 4 + 3 x 2 + 4x + 7 h(f(3)) h(3 + 2) h(5) 5 2 + 3 25 + 3 = 28 f(x) = x + 2h(x) = x 2 + 3

20. Goal - Find inverses of linear functions. State Standard 24.0 – Students solve problems involving inverse functions STEP 1Switch the “y” and the “x” values. Solving for the Inverse STEP 2Solve for “y”.

21. Example 1: Find the inverse of 10y +2x = 4 Answer: y -1 = -5x + 2 10x + 2y = 4 2y = -10x + 4 y = -5x + 2 2 2

22. Example 2: Find the inverse of y = -3x + 6 Answer: y -1 = (-1/3)x + 2 x = -3y + 6 x – 6 = -3y –6 –3 y = (-1/3)x + 2

23. f -1 (x)= 5 x Example 3: Find the inverse of the function: f(x) = x 5 x = y 5 5 x = y y = x 5

24. 1) Identify the domain and range InputOutput 2 5 6 3 -2 2) Graph y = -2x 2 + 3 Domain = -1, 2, 5 & 6 Range = -2 & 3

25. x y x y Domain: x > 0, Range: y > 0Domain and range: all real numbers (0,0) (1,1) (0,0) (-1,-1) Objective- Students will learn to graph functions of the form y = a x – h + k and y = a 3 x – h + k.

26.  Example 1 Describe how to create the graph of y = x + 2 – 4 from the graph of y = x. Comparing Two Graphs Solution h = -2 and k = -4 shift the graph to the left 2 units & down 4 units

27. Graphs of Radical Functions To graph y = a x - h + k or y = a 3 x - h + k, follow these steps. STEP  Sketch the graph of y = a x or y = a 3 x. STEP  Shift the graph h units horizontally and k units vertically

28.  Example 2 Graph y = -3 x – 1 + 3. Graphing a Square Root Solution 1)Sketch the graph of y = -3 x (dashed). It begins at the origin and passes through point (1,-3). (1, 3) (1,-3) (0,0)(2,0) 2) For y = -3 x – 1 + 3, h = 1 & k = 3. Shift both points 1 to the right and 3 up.

29.  On White Board Graph y = 2 x – 2 + 1. Graphing a Square Root (2, 1) (1,2) (0,0) (3,3)

30. Graph y = 2 3 x + 3 – 4.  Example 3 Graphing a Cube Root Solution 1)Sketch the graph of y = 2 3 x (dashed). It passed through the origin & the points (1, 2) & (-1, -2). (-4,-6) (0,0) (-3,-4) (-2,-2) (-1,-2) (1, 2) 2) For y = 2 x + 3 – 4, h = -3 & k = -4. Shift the three points Left 3 and Down 4.

31. Graph y = 3 3 x – 2 + 1  On White Board Graphing a Cube Root (1,-2) (0,0) (2,1) (3,4) (-1,-3) (1, 3)