1. Objectives Graph radical functions and inequalities.
Transform radical functions by changing parameters.

2. Recall that exponential and logarithmic functions are inverse functions. Quadratic and cubic functions have inverses as well. The graphs below show the inverses of the quadratic parent function and cubic parent function.

3. Notice that the inverses of f(x) = x2 is not a function because it fails the vertical line test. However, if we limit the domain of f(x) = x2 to x ≥ 0, its inverse is the function
A radical function is a function whose rule is a radical expression. A square-root function is a radical function involving . The square-root parent function is The cube-root parent function is

4. Example 1A: Graphing Radical Functions
Graph each function and identify its domain and range.
Make a table of values. Plot enough ordered pairs to see the shape of the curve. Because the square root of a negative number is imaginary, choose only nonnegative values for x – 3.

5. x (x, f(x)) Example 1A Continued 3 (3, 0) 4 (4, 1) 7 (7, 2) 12 (12, 3)● ● ● ●
The domain is {x|x ≥3}, and the range is {y|y ≥0}.

6. Example 1B: Graphing Radical Functions
Graph each function and identify its domain and range.
Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

7. x (x, f(x)) Example 1B Continued –6 (–6, –4) 1 (1,–2) 2 (2, 0) 3
(3, 2) 10
(10, 4) ● ● ● ● ●
The domain is the set of all real numbers. The range is also the set of all real numbers

8. Example 1B Continued
Check Graph the function on a graphing calculator.

9. Check It Out! Example 1a
Graph each function and identify its domain and range.
Make a table of values. Plot enough ordered pairs to see the shape of the curve. Choose both negative and positive values for x.

10. Check It Out! Example 1a Continued
(x, f(x)) –8
(–8, –2) –1
(–1,–1)
(0, 0) 1
(1, 1) 8
(8, 2) • • • • •
The domain is the set of all real numbers. The range is also the set of all real numbers.

11. Check It Out! Example 1a Continued
Check Graph the function on a graphing calculator.

12. • • • • x (x, f(x)) Check It Out! Example 1b
Graph each function, and identify its domain and range. x
(x, f(x)) –1
(–1, 0) 3
(3, 2) 8
(8, 3) 15
(15, 4) • • • •
The domain is {x|x ≥ –1}, and the range is {y|y ≥0}.

13. The graphs of radical functions can be transformed by using methods similar to those used to transform linear, quadratic, polynomial, and exponential functions. This lesson will focus on transformations of square-root functions.

15. Example 2: Transforming Square-Root Functions
Using the graph of as a guide, describe the transformation and graph the function.
f(x) = x
g(x) = x + 5 •
Translate f 5 units up.

16. Check It Out! Example 2a
Using the graph of as a guide, describe the transformation and graph the function.
f(x)= x •
g(x) = x + 1
Translate f 1 unit up.

17. g is f vertically compressed by a factor of .
Check It Out! Example 2b
Using the graph of as a guide, describe the transformation and graph the function.
f(x) = x
g is f vertically compressed by a factor of 1 2

18. Transformations of square-root functions are summarized below.

19. Example 3: Applying Multiple Transformations
Using the graph of as a guide, describe the transformation and graph the function
f(x)= x . •
Reflect f across the x-axis, and translate it 4 units to the right.

20. Check It Out! Example 3a
Using the graph of as a guide, describe the transformation and graph the function.
f(x)= x ●
g is f reflected across the y-axis and translated 3 units up.

21. Check It Out! Example 3b
Using the graph of as a guide, describe the transformation and graph the function.
f(x)= x ●
g(x) = –3 x – 1
g is f vertically stretched by a factor of 3, reflected across the x-axis, and translated 1 unit down.

22. Example 4: Writing Transformed Square-Root Functions
Use the description to write the square-root function g. The parent function is reflected across the x-axis, compressed vertically by a factor of , and translated down 5 units.
f(x)= x 1 5
Step 1 Identify how each transformation affects the function.
Reflection across the x-axis: a is negative 1 5
a = –
Vertical compression by a factor of 1 5
Translation 5 units down: k = –5

23. Step 2 Write the transformed function.
Example 4 Continued
Step 2 Write the transformed function. 1 5
g(x) = x + (- 5) ö ÷ ø æ ç è
Substitute – for a and –5 for k. 1 5
Simplify.

24. Example 4 Continued
Check Graph both functions on a graphing calculator. The g indicates the given transformations of f.

25. Check It Out! Example 4
Use the description to write the square-root function g.
The parent function is reflected across the x-axis, stretched vertically by a factor of 2, and translated 1 unit up.
f(x)= x
Step 1 Identify how each transformation affects the function.
Reflection across the x-axis: a is negative
a = –2
Vertical compression by a factor of 2
Translation 5 units down: k = 1

26. Check It Out! Example 4 Continued
Step 2 Write the transformed function.
Substitute –2 for a and 1 for k.
Simplify.
Check Graph both functions on a graphing calculator. The g indicates the given transformations of f.

27. Example 5: Business Application
A framing store uses the function to determine the cost c in dollars of glass for a picture with an area a in square inches. The store charges an addition $6.00 in labor to install the glass. Write the function d for the total cost of a piece of glass, including installation, and use it to estimate the total cost of glass for a picture with an area of 192 in2.

28. Example 5 Continued
Step 1 To increase c by 6.00, add 6 to c.
Step 2 Find a value of d for a picture with an area of 192 in2.
Substitute 192 for a and simplify.
The cost for the glass of a picture with an area of 192 in2 is about $13.13 including installation.

29. Check It Out! Example 5
Special airbags are used to protect scientific equipment when a rover lands on the surface of Mars. On Earth, the function f(x) = approximates an object’s downward velocity in feet per second as the object hits the ground after bouncing x ft in height. 64 x
The downward velocity function for the Moon is a horizontal stretch of f by a factor of about Write the velocity function h for the Moon, and use it to estimate the downward velocity of a landing craft at the end of a bounce 50 ft in height. 25 4

30. Check It Out! Example 5 Continued
Step 1 To compress f horizontally by a factor of , multiply f by 25 4 4 25 4 25
h(x) = f x = · 64x = æ ç è ö ÷ ø
Step 2 Find the value of g for a bounce of 50ft.
h (x) = · »
256 23 25 50
Substitute 50 for x and simplify.
The landing craft will hit the Moon’s surface with a downward velocity of about 23 ft at the end of the bounce.

31. In addition to graphing radical functions, you can also graph radical inequalities. Use the same procedure you used for graphing linear and quadratic inequalities.

32. Example 6: Graphing Radical Inequalities
Graph the inequality
Step 1 Use the related equation to make a table of values.
y =2 x -3 x 1 4 9 y –3 –1 3

33. Example 6 Continued
Step 2 Use the table to graph the boundary curve. The inequality sign is >, so use a dashed curve and shade the area above it.
Because the value of x cannot be negative, do not shade left of the y-axis.

34. Example 6 Continued
Check Choose a point in the solution region, such as (1, 0), and test it in the inequality.
0 > 2(1) – 3
0 > –1 

35. x –4 –3 5 y 1 2 3 Check It Out! Example 6a Graph the inequality.
Step 1 Use the related equation to make a table of values.
y = x+4 x –4 –3 5 y 1 2 3

36. Check It Out! Example 6a Continued
Step 2 Use the table to graph the boundary curve. The inequality sign is >, so use a dashed curve and shade the area above it.
Because the value of x cannot be less than –4, do not shade left of –4.

37. Check It Out! Example 6a Continued
Check Choose a point in the solution region, such as (0, 4), and test it in the inequality.
4 > (0) + 4
4 > 2 

38. x –4 –3 5 y 1 2 3 Check It Out! Example 6b Graph the inequality.
Step 1 Use the related equation to make a table of values. 3
y = x - 3 x –4 –3 5 y 1 2 3

39. Check It Out! Example 6b Continued
Step 2 Use the table to graph the boundary curve. The inequality sign is >, so use a dashed curve and shade the area above it.

40. Check It Out! Example 6b Continued
Check Choose a point in the solution region, such as (4, 2), and test it in the inequality.
2 ≥ 1 

41. Lesson Quiz: Part I
1. Graph the function and identify its range and domain.
D:{x|x≥ –4}; R:{y|y≥ 0} •

42. Lesson Quiz: Part II
2. Using the graph of as a guide, describe the transformation and graph the function
g(x) = -x + 3
g is f reflected across the y-axis and translated 3 units up. •

43. Lesson Quiz: Part III
3. Graph the inequality