We have come up with the following definitions.
-
x0=1
x
0
1
-
x−a=
1xa
x
a
=1xa size 12{ { {1} over {x rSup { size 8{a} } } } } {}
-
xab
=xab
x
a
b
=xab size 12{ nroot { size 8{b} } {x rSup { size 8{a} } } } {}
Let’s get a bit of practice using these definitions.
Check all of your answers above on your calculator. If any of them did not come out right, figure out what went wrong, and fix it!
Solve for
x
x:
xx1712
xx size 12{ { {x rSup { size 8{ {3} wideslash {2} } } } over {x rSup { size 8{ {1} wideslash {2} } } } } } {}
17
1
2
1712
17
1
2
Solve for
x
x:
x12=9
x
1
2
9
Simplify:
xxxx size 12{ { {x} over { sqrt {x} } } } {}
Simplify:
x+xx+1xx+xx+1x size 12{ { {x rSup { size 8{ {3} wideslash {2} } } + sqrt {x} } over {x rSup { size 8{ {5} wideslash {2} } } + { {1} over { sqrt {x} } } } } } {}
Multiply the top and bottom by
x12
x
1
2
.
Now…remember inverse functions? You find them by switching the
x
x and the
y
y and then solving for
y
y. Find the inverse of each of the following functions. To do this, in some cases, you will have to rewrite the things. For instance, in #9, you will start by writing
y=x12
y
x
1
2
. Switch the
x
x and the
y
y, and you get
x=y12
x
y
1
2
. Now what? Well, remember what that means: it means
x=y
x
y
. Once you’ve done that, you can solve for
y
y, right?
x3
x
3
- a. Find the inverse function.
- b. Test it.
x-2
x
-2
- a. Find the inverse function.
- b. Test it.
x0
x
0
- a. Find the inverse function.
- b. Test it.
Can you find a generalization about the inverse function of an exponent?
Graph
y=2x
y
2
x
by plotting points. Make sure to include both positive and negative xx values.
Graph
y=2×2x
y
2
2
x
by doubling all the y-values in the graph of
y=2x
y
2
x
.
Graph
y=2x+1
y
2
x
1
by taking the graph
y=2x
y
2
x
and “shifting” it to the left by one.
Graph
y=12x
y
1
2
x
by plotting points. Make sure to include both positive and negative xx values.
"This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them in […]"