1. Section 4.4
Radical Functions

2. Objectives:
1. To restrict the domain of a
function that is not one-to-one so
that the inverse is a function.
2. To graph radical functions and
give their domains.

3. Definition
Radical function A radical function is a function containing at least one variable in the radicand.

4. The most common radical function is the square root function
The most common radical function is the square root function. All even root functions have both a positive and a negative root, but only the principal (positive) root is used. A square root function, whose simplest form is
f(x) = x indicates only the principal square root.

5. x y x
f(x) = x 1 2 3 4 7 1
2  1.4
3  1.7 2
7  2.6

6. EXAMPLE 1 Find the function described by the rule h(x) = x + 3 where the domain is {0, -3, 1, 4, 6}.

7. EXAMPLE 1 Find the function described by the rule h(x) = x + 3 where the domain is {0, -3, 1, 4, 6}.

8. Rational functions require restrictions so the denominator is not zero
Rational functions require restrictions so the denominator is not zero. Radical functions require the radicand to be nonnegative.

9. Find the domain of the following function.2 x 3 6 ) ( g + = 2 x 3 + 2 x 3 - 3 2 x -  þ ý ü î í ì -  = 3 2 x D g

10. Find the domain of the following function.3 4 x ) ( p - =
Dp = all real numbers
Since the index (3) is odd, no restrictions are needed.

11. EXAMPLE 2 Find the domain for each of the following functions.
a. g(x) = – 3x – 7, b. q(x) = 5x + 7,
c. r(x) =
2x + 4
(x – 2) x - 1 3

12. EXAMPLE 2 Find the domain for each of the following functions.
a. g(x) = – 3x – 7
a. for g(x) 4 – 3x  0
-3x  -4
x  4 3
Dg = {x|x  } 4 3

13. EXAMPLE 2 Find the domain for each of the following functions.
b. q(x) = 5x + 7 3
b. for Q(x) Dq = {all real numbers}

14. EXAMPLE 2 Find the domain for each of the following functions.
c. r(x) =
2x + 4
(x – 2) x - 1
c. for r(x) x – 1  0, x – 2  0
x  1 x  2
Dr = {x|x  1 and x  2}

15. EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4
EXAMPLE 3 Find the inverse relation of f(x) = x If the inverse is not a function, then restrict the domain so that it becomes one.
y = x2 + 4
x = y2 + 4
y2 = x – 4
y = ± x – 4
f-1(x) = ± x – 4

16. EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4
EXAMPLE 3 Find the inverse relation of f(x) = x If the inverse is not a function, then restrict the domain so that it becomes one.
If the domain of f(x) is restricted to x  0 then f(x) would be one-to-one and the function would have an inverse.

17. EXAMPLE 3 Find the inverse relation of f(x) = x2 + 4
EXAMPLE 3 Find the inverse relation of f(x) = x If the inverse is not a function, then restrict the domain so that it becomes one.
If only the blue portion of f is used, then f-1(x) = x – 4. The blue portion of f-1 is the inverse function.

18. EXAMPLE 4 Find the inverse function for f(x) = - x – 4.
y = - x – 4
x = - y – 4
x2 = (- y – 4)2
x2 = y – 4
x2 + 4 = y
f-1(x) = x2 + 4, x  0

19. Homework pp

20. ►A. Exercises
Give the domain of each. Also identify the x-intercept and the y-intercept.
7. q(x) = x + 1

21. ►A. Exercises
Give the domain of each. Also identify the x-intercept and the y-intercept.
9. r(x) = x + 2 3

22. ►A. Exercises
Give the domain of each. Also identify the x-intercept and the y-intercept.
11. f(x) = 2x – 5

23. ►B. Exercises
Graph each radical function. Identify the domain and range.
13. g(x) = -x

24. ►B. Exercises
For each graph below, find a one-to-one portion of the graph (restrict the domain), and then sketch the inverse. Write the restricted domains of f(x) and f-1(x) in interval notation.

25. ►B. Exercises
19.

26. ►B. Exercises
21.

27. ■ Cumulative Review Find the areas of the following triangles.
26. with sides of 29, 36, and 47.

28. ■ Cumulative Review Find the areas of the following triangles.
27. with A = 37°, b = 12.5, and c = 17.0

29. ■ Cumulative Review Find the areas of the following triangles.
28. with a leg of 25 and a hypotenuse of 47.

30. ■ Cumulative Review
29. Which two trig functions have a period of ?

31. ■ Cumulative Review
30. What type of symmetry does an odd function have?