Radical Functions,

1
Chapter 5 Functions
5.7
Radical Functions, Equations, and Inequalities
5.7.1
MATHPOWERTM 11, WESTERN EDITION
2
Radical Equations
An equation in which a variable occurs in the
radicand is called a radical equation. It should
be noted, that when solving a radical equation
algebraically, extraneous roots may be
introduced when both sides of an equation are
squared. Therefore, you must check your
solutions for a radical equation.
Check
L.S. R.S.
Solve v x - 3 - 3 0
x 3
0
v x - 3 - 3
v x - 3 3
(v x - 3 )2 (3)2
v 12 - 3 - 3
3 - 3 0
x - 3 9 x 12
Therefore, the solution is x 12.
5.7.2
3
Solving Radical Equations
4 v 4 x2 x
Check
x
v 4 x2 x - 4
(v 4 x2)2 (x - 4)2
4 x2 x2 - 8x 16 8x 12
Since
the solution of
is extraneous. Therefore, there are
no real roots.
x
?
5.7.3
4
Solving Radical Equations
x -2
Solve
Set up the equation so that there will be one
radical on each side of the equal sign.
Square both sides.
2x 4 x 7 x 3
Simplify.
L.S. R.S.
Verify your solution.
Therefore, the solution is x 3.
5.7.4
5
Squaring a Binomial
Note that the middle term is twice the product
of the two terms of the binomial.
(a 2)2 a2 4a 4
( 5 vx - 2 )2
The middle term will be twice the product of
the two terms.
A final concept that you should know
(avx b)2
a2(x b)
a2x ab
5.7.5
6
Solving Radical Equations
Set up the equation so that there will be only
one radical on each side of the equal sign.
Solve
Square both sides of the equation.
Use Foil.
Simplify.
Simplify by dividing by a common factor of 2.
Square both sides of the equation.
Use Foil.
5.7.6
7
Solving Radical Equations
Distribute the 4.
Simplify.
Factor the quadratic.
Solve for x.
x - 3 0 or x - 7 0 x 3 or
x 7
Verify both solutions.
L.S. R.S.
L.S. R.S.
5.7.7
8
Graphing a Radical Function
Graph
The domain is x gt -2. The range is y gt 0.
5.7.8
9
Solving a Radical Equation Graphically
The solution will be the intersection of the
graph
Solve
and the graph of y 0.
The solution is x 12.
L.S. R.S.
Check
0
5.7.9
10
Solving a Radical Equation Graphically
Solve
The solution is x 3 or x 7.
5.7.10
11
Solving Radical Inequalities
Solve
Find the values for which the graph of
Note the radical 7x - 3 is defined only when
.
is above the graph of y 3.
The graphs intersect at x 4.
x gt 4
Therefore, the solution is x gt 4.
5.7.11
12
Solving Radical Inequalities
x gt -1
Solve
The graphs intersect at the point where x 8.
x -1 and x lt 8
The solution is -1 lt x and x lt 8.
5.7.12
13
Solving Radical Inequalities
x -3
Solve
x ? -2
The solution is -3 lt x lt -2 and
x gt - 1.2.
5.7.13
14
Assignment
Suggested Questions
Pages 323-326 1, 5, 9, 13, 17, 21, 23, 27, 31,
35, 39, 43, 45, 59
69, 74, 103ad, 104ac, 119, 126, 132ghi