Download File - Mrs. Hille`s FunZone

Section 4.4
Radical Functions
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.
Definition
Radical function A radical
function is a function containing at
least one variable in the radicand.
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.
x f(x) = x
0
0
1
1
2  1.4
2
3  1.7
3
4
2
7
7  2.6
y
x
EXAMPLE 1 Find the function
described by the rule h(x) = x + 3 where
the domain is {0, -3, 1, 4, 6}.
h(0) = 0 + 3 = 3
h(-3) = -3 + 3 = 0
h(1) = 1 + 3 = 2
h(4) = 4 + 3 = 7
h(6) = 6 + 3 = 3
h = {(-3, 0), (0, 3), (1, 2), (4, 7), (6, 3)}
EXAMPLE 1 Find the function
described by the rule h(x) = x + 3 where
the domain is {0, -3, 1, 4, 6}.
h = {(-3, 0), (0, 3), (1, 2), (4, 7), (6, 3)}
Rational functions require
restrictions so the
denominator is not zero.
Radical functions require the
radicand to be nonnegative.
Find the domain of the
following function.
g(x ) = 6 3x + 2
3x + 2  0
3x  - 2
-2
x
3
- 2

Dg =  x x 


3 
Find the domain of the
following function.
3
p( x ) = x - 4
Dp = all real numbers
Since the index (3) is odd, no
restrictions are needed.
EXAMPLE 2 Find the domain for
each of the following functions.
3
a. g(x) = 3 4 – 3x – 7, b. q(x) = 5x + 7,
2x + 4
c. r(x) =
(x – 2) x - 1
EXAMPLE 2 Find the domain for
each of the following functions.
a. g(x) = 3 4 – 3x – 7
a. for g(x) 4 – 3x  0
-3x  -4
4
x
3
4
Dg = {x|x  }
3
EXAMPLE 2 Find the domain for
each of the following functions.
3
b. q(x) = 5x + 7
b. for Q(x)
Dq = {all real numbers}
EXAMPLE 2 Find the domain for
each of the following functions.
2x + 4
c. r(x) =
(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}
EXAMPLE 3 Find the inverse
relation of f(x) = x2 + 4. 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
EXAMPLE 3 Find the inverse
relation of f(x) = x2 + 4. 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.
EXAMPLE 3 Find the inverse
relation of f(x) = x2 + 4. 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.
EXAMPLE 4 Find the inverse
function for f(x) = - x – 4.
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
Homework
pp. 191-193
►A. Exercises
Give the domain of each. Also identify the
x-intercept and the y-intercept.
7. q(x) = x + 1
►A. Exercises
Give the domain of each. Also identify the
x-intercept and the y-intercept.
3
9. r(x) = x + 2
►A. Exercises
Give the domain of each. Also identify the
x-intercept and the y-intercept.
5
11. f(x) = 2x – 4 + 7.75
►B. Exercises
Graph each radical function. Identify the
domain and range.
13. g(x) = -x
►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.
►B. Exercises
19.
►B. Exercises
21.
■ Cumulative Review
Find the areas of the following triangles.
26. with sides of 29, 36, and 47.
■ Cumulative Review
Find the areas of the following triangles.
27. with A = 37°, b = 12.5, and c = 17.0
■ Cumulative Review
Find the areas of the following triangles.
28. with a leg of 25 and a hypotenuse
of 47.
■ Cumulative Review
29. Which two trig functions have a
period of ?
■ Cumulative Review
30. What type of symmetry does an
odd function have?