54
Solving Radical Equations (Section 3.4)
Example. Simplify the following.
p
64
p
4
81
p
3
p
3
27
p
5
p
4
32
How do you use your calculator to help with things like
p
4
8
1
15?
Example. Use your calculator to round the following to three decimal places.
p
3
83
p
4
76
p
5
19
55
How do we solve an equation like
p
3
2
x = 4?
Example.
Solve each of the following equations.p
p
4x + 1 = 3
y
p
5
3x + 4 = 2
p
3
6x
1+4=0
9+8=5
56
How do we solve an equation like
p
x + 1 + 1 = x?
Example.
Solve each of the following equations. p
p
7x + 4 = x + 2
x
3+5=x
57
Graphs of Rational Functions (Section 4.5)
Definition. A rational function if a function of the form:
where p(x) and q(x) are polynomials and q(x) 6= 0.
Example. Graph the following functions on your graphing calculator and determine the domain.
3
x+1
f (x) =
f (x) = 2
x+5
x
x 6
Remark. The domain for a rational function f (x) =
p(x)
is:
q(x)
Example. Determine the domain for each of the following functions.
2x 11
x2 4x
f (x) = 2
f (x) = 3
x + 2x 8
x
x
58
Graph f (x) =
x2
x3
4x
on your graphing calculator:
x
-9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9
p(x)
:
q(x)
(1) Reduce f (x) by canceling out any common factors between p(x) and q(x).
(2) Set the new denominator equal to zero.
(3) The vertical asymptotes correspond to the answers found above.
Remark. To find vertical asymptotes for a rational function f (x) =
Example. Determine the vertical asymptotes for each of the following rational functions.
x2 2x
2x + 6
f (x) = 2
f (x) = 2
x + 2x 8
x + 7x 8
59
Solving Rational Equations (Section 3.4)
How do we solve an equation like
x
8
3
+
. . . or how would you solve an equation like
x
3
2
= 0?
2x
5
=
?
x+7
x 3
60
Example. Solve the following equations.
6
x+ =5
x
2
x
1
=
3
x+2
1 2
1 3
+ = +
2 x
3 x
3y + 5
y+4
y+1
+
=
2
y + 5y y + 5
y
61
Compositions of Functions (Section 2.3)
In the beginning of the semester, we did problems like this one:
For f (x) = 3x
f (4 + h)
1 and g(x) = x2 + 4x
1 find the following:
g(1 + h)
This leads into the composition of functions. For the two functions f (x) and g(x), the composition functions f g and g f are defined as:
Example. Consider the functions f (x) = 3x + 1 and g(x) = x2 + 4. Find:
(f g)(x)
(g f )(x)
Remark. Notice that: