1. Describe the end behavior of f (x) = 4x 4 + 2x – 8.
5–Minute Check 2

2. Identify the parent function f (x) of g (x) = 2|x – 3| + 1
Identify the parent function f (x) of g (x) = 2|x – 3| + 1. Describe how the graphs of g (x) and f (x) are related.
A. f (x) = | x |; f (x) is translated 3 units right, 1 unit up and expanded vertically to graph g (x).
B. f (x) = | x |; f (x) is translated 3 units right, 1 unit up and expanded horizontally to graph g (x).
C. f (x) = | x |; f (x) is translated 3 units left, 1 unit up and expanded vertically to graph g (x).
D. f (x) = | x |; f (x) is translated 3 units left, 1 unit down and expanded horizontally to graph g (x).
5–Minute Check 3

3. Find [f ○ g](x) and [g ○ f ](x) for f (x) = 2x – 4 and g (x) = x 2.
A. (2x – 4)x 2; x 2(2x – 4)
B. 4x 2 – 16x + 16; 2x 2 – 4
C. 2x 2 – 4; 4x 2 – 16x + 16
D. 4x 2 – 4; 4x
5–Minute Check 4

4. Evaluate f (2x) if f (x) = x 2 + 5x + 7.
A. 2x x + 7
B. 2x x 2 + 7
C. 4x x + 7
D. 4x 2 + 7x + 7
5–Minute Check 5

5. power function monomial function radical function extraneous solution
Vocabulary

6. Key Concept 1

7. Analyze Monomial Functions
A. Graph and analyze Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.
Evaluate the function for several x-values in its domain. Then use a smooth curve to connect each of these points to complete the graph.
Example 1

8. continuity: continuous for all real numbers;
Analyze Monomial Functions
D = (–∞, ∞); R = [0, ∞);
intercept: 0;
end behavior:
continuity: continuous for all real numbers;
decreasing: (–∞, 0); increasing: (0, ∞)
Example 1

9. Answer: D = (–∞, ∞); R = [0, ∞); intercept: 0;
Analyze Monomial Functions
Answer: D = (–∞, ∞); R = [0, ∞); intercept: 0;
continuous for all real numbers; decreasing: (–∞, 0) , increasing: (0, ∞)
Example 1

10. Functions with Negative Exponents
A. Graph and analyze f (x) = 2x – 4. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.
Example 2

11. continuity: infinite discontinuity at x = 0;
Functions with Negative Exponents
intercept: none;
end behavior:
continuity: infinite discontinuity at x = 0;
increasing: (–∞, 0); decreasing: (0, ∞)
Example 2

12. Answer: D = (– ∞, 0)  (0, ∞); R = (0, ∞); no intercept ;
Functions with Negative Exponents
Answer: D = (– ∞, 0)  (0, ∞); R = (0, ∞); no intercept ;
infinite discontinuity at x = 0; increasing: (–∞, 0), decreasing: (0, ∞);
Example 2

13. Functions with Negative Exponents
B. Graph and analyze f (x) = 2x –3. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.
Example 2

14. D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞);
Functions with Negative Exponents
D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞);
intercept: none;
end behavior:
continuity: infinite discontinuity at x = 0;
decreasing: (–∞, 0) and (0, ∞)
Example 2

15. Functions with Negative Exponents
Answer: D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞); no intercept ; infinite discontinuity at x = 0; decreasing: (–∞, 0) and (0, ∞)
Example 2

16. Describe the end behavior of the graph of f (x) = 3x –5.
Example 2

17. Rational Exponents
A. Graph and analyze Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.
Example 3

18. continuity: continuous on [0, ∞);
Rational Exponents
D = [0, ∞); R = [0, ∞);
intercept: 0;
end behavior:
continuity: continuous on [0, ∞);
increasing: [0, ∞)
Example 3

19. Rational Exponents
Answer: D = [0, ∞); R = [0, ∞); intercept: 0; ; continuous on [0, ∞); increasing: [0, ∞)
Example 3

20. Rational Exponents
B. Graph and analyze Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.
Example 3

21. continuity: continuous on (0, ∞);
Rational Exponents
D = (0, ∞); R = (0, ∞);
intercept: none;
end behavior:
continuity: continuous on (0, ∞);
decreasing: (0, ∞)
Example 3

22. Rational Exponents
Answer: D = (0, ∞); R = (0, ∞); no intercept ; continuous on (0, ∞); decreasing: (0, ∞)
Example 3

23. Describe the continuity of the function .
A. continuous for all real numbers
B. continuous on and
C. continuous on (0, ∞]
D. continuous on [0, ∞)
Example 3

24. Key Concept 5

25. Graph Radical Functions
A. Graph and analyze Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.
Example 5

26. Graph Radical Functions
Answer: D = [0, ∞); R = [0, ∞); intercept: 0; ; continuous on [0, ∞); increasing: [0, ∞)
Example 5

27. Graph Radical Functions
B. Graph and analyze Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing.
Example 5

28. Graph Radical Functions
Answer: D = (–∞, ∞) ; R = (–∞, ∞) ; x-intercept: , y-intercept: about –0.6598; ; continuous for all real numbers; increasing: (–∞, ∞)
Example 5

29. Find the intercepts of the graph of .
A. x-intercept: , y-intercept:
B. x-intercepts: , y-intercept:
C. x-intercept: , y-intercept:
D. x-intercepts: , y-intercept –4
Example 5

30. Square each side to eliminate the radical.
Solve Radical Equations
A. Solve
original equation
Isolate the radical.
Square each side to eliminate the radical.
Subtract 28x and 29 from each side.
Factor.
Factor.
x – 5 = 0 or x + 1 = 0
Zero Product Property
x = 5 x = –1
Solve.
Example 6

31. Answer: –1, 5 Check x = 5 x = –1 10 = 10  –2 = –2 
Solve Radical Equations
Answer: –1, 5
Check
x = 5
x = –1
10 = 10 
–2 = –2 
A check of the solutions in the original equation confirms that the solutions are valid.
Example 6

32. Subtract 8 from each side.
Solve Radical Equations
B. Solve
original equation
Subtract 8 from each side.
Raise each side to the third power. (The index is 3.)
Take the square root of each side.
x = 10 or –6
Add 2 to each side.
A check of the solutions in the original equation confirms that the solutions are valid.
Answer: 10, –6
Example 6

33. Distributive Property Combine like terms. (x – 8)(x – 24) = 0 Factor.
Solve Radical Equations
C. Solve
original equation
Square each side.
Isolate the radical.
Square each side.
Distributive Property
Combine like terms.
(x – 8)(x – 24) = 0
Factor.
x – 8 = 0 or x – 24 = 0
Zero Product Property
Example 6

34. Solve Radical Equations
x = 8 x = 24
Solve.
One solution checks and the other solution does not. Therefore, the solution is 8.
Answer: 8
Example 6

35. Solve
A. 0, 5
B. 11, –11
C. 11
D. 0, 11
Example 6