I f-1(x) or Inverse
1. Write equation as y =
2. Replace every x with y and y’s with x’s
3. Solve for y.
Mult choice: type in f(x) look at table. Then type
in each answer choice to see if the x’s and y’s
have switched.
0
3
3
0
1
5
5
1
2
7
7
2
**Also: f(x) and g(x) are inverses if if fog(x) = x
Example: f(x)= 2x – 4. Find f-1(x)
y = 2x – 4  x = 2y – 4  x + 4 = 2y 
x4 1
or x  2
2
2
=y
x4 1
or x  2
2
f (x) = 2
-1
II f o g (x) or f(g(x))
Plug g(x) into f(x)&simplify
f o g(-2) Substitute (-2) into g, get answer,
substitute answer into f
USE ( ) for negative #’s and fraction
Ex: f(x) = x2 – 3x g(x) = (x – 2)
f o g(x) = (x – 2)2 – 3x
= (x – 2)(x – 2) - 3(x – 2)
= x2 – 4x + 4 – 3x + 6
= x2 – 7x + 10
f o g(-2) = f ( g(-2) ) = f ( -4) = (-4)2 – 3(-4) =16 + 12 = 28
III Applications of f(g(x)
Taxes after discount.
Tips after taxes.
IV Vertical asymptotes
Type into y=. Use( ) on top &( ) on bottom.
Look at graph for
.
There will be ERROR in the table also.
V Holes
Type into y=. Use ( ) on top and ( ) on bottom.
There will be an ERROR on the calculator but
there will NOT BE A VERTICAL ASYMPTOTE
VI Horizontal Asymptotes
1) Go to Tableset 2nd Window
2) Change TbleStart = 10000
3) Go to Table 2nd Graph
4) Look at y’s.
It will be *close to 0 (or E-4), OR
*close to a number,
OR
*NONE as the graph keeps getting larger
1) g(x) =
a) 3x – 5
2
x5
3
.
Find g-1(x)
b) 3x – 15 c) 1.5x – 7.5
2) h(x)= x3 - 5. Find h-1(x)
x5
3
a) x  5 b) x3 + 5 c) 3 d)
3
x 5
3) What is true about the composition of two
functions f(x), g(x) if they are inverses of each
other?
a) f(g(x))= g(f(x))= x
b) f(g(x)) = g(f(x)) = 0
c) f(g(x)) = g(f(x))=1
d) f(g(x)) = g(f(x)) = 1/x
2
f(x) = 2x – 7 g ( x)  x  4
3
h(x)= x  5
4) What is f(h(3))?
a) -6 b) 19 c) 37 d) -15
5) What is (g o f)(x)?
3
2
a) 2 x  7 x  8 x  28
b) 2 x 2  1
c) 2 x 4  7 x 3  8 x 2  28 x d) 4 x 2  28 x  53
6) Shoes are discounted, d(x), at 20% off. Sales
tax, s(x), is 8%. Jane buys a $50 pair of shoes.
Find (s o d)(x)
No multiple choice
7) Find the vertical asymptotes of
x3
2
f(x) = x  5 x  6
a) x = 3 b) x = 2 c) x = 3, x = 2 d) None
x3
2
x

5x  6
8) Find the hole(s) of f(x) =
a) x = 3 b) x = 2 c) x = 3, x = 2 d) None
x2
9) Find the hole(s) of g(x) = x  8 x  12
2
a) x = 6 b) x = 4 c) None
d) x = 2
10) Find the horizontal asymptote of
x3
f(x) = x  5 x  6 a) y = 0
b) None c) y = 1
11) Find the horizontal asymptote of
2
3x 2  3
2 x 2  5x  6
f(x) =
a) y = 0
b) None c) y = 1
12) Find the horizontal asymptote of
h(x)=
x2  3
5x  6
a) y = 0
b) None c) y = 1
d) y = ½
d) y = 3/2
d) y = 1/5
VII Direct Variation
1) Make equation y=kx.
2) Plug into the equation twice.
3) Solve for k in one of them and substitute into the other
one.
Square of x: x2
Square root of x:
Cube of x: x3
Cube root of x:
3
x
x
Ex: y varies directly as square of x. x is 3 when y is 36.
Find y when x is 10.
1) y = kx2
2) 36 = k(3)2
y = k(10)2
9=k
y = 9(10)2 = 900
VIII Inverse Variation
k
1) Make equation y= x
2) Plug into the equation twice
3) Solve for k in one of them and substitute into the other
one.
Example: y varies inversely as square as x. x is 3 and y is
7. Find y when x is 4.
k
2
1) y= x
k
2
2) 7 = 3
63 = k
y=
k
2
y=4
63
16
IX Joint Variation
It is similar to direct variation with more
variables. Put any “inverses” on the bottom.
X Deciding if it’s INVERSE or DIRECT
*If one increases makes the other one decrease
then its INVERSE
Ex: speed/time; tractors/time; painters/time
*If both increases or both decreases then its
DIRECT
Ex: $/hours worked; Volume/radius
XI Square Roots
1) 5 STO X Enter
2) 7 STO ALPHA 1 Enter
Type in original equation and write down number
Type in each mult. choice and look for same
number
Ex:
25 x 2 y is 66.1438. 5x y is also 66.1438
13) If r varies directly as the cube root of s, and r=32 when s
= 8, find r when s = 64
a) 4 b) 16 c) 64 d) 256
14) A car needs 35 feet to come to a stop after the brakes are
applied at 30mph. The braking distance, in feet, varies
directly as the square of the speed, in miles per hour. To the
nearest foot, what is the breaking distance for this car when
it is traveling 50 mph?
a) 55 ft. b) 58 ft. c) 83 ft. d) 97 ft.
15) If y varies inversely as the cube of x, and y = 10 when x
= 4, find the value of y when x = 2.
a) 1.25 b) 5 c) 80 d) 500
16) Write a single equation for this situation:
p varies directly as q and inversely as the square root of r.
k r
kq
kq
kq
2
a) p= r
b) p= r c) p= q d)p= r
17) The weight of a body on Earth’s surface varies inversely
as the square of the distance from Earth’s center. Earth’s
radius is about 4,000 miles. If an astronaut weighs 185
pounds on Earth’s surface, how much will he weigh in orbit
22,000 miles above Earth’s surface?
a) 4.4 lbs b) 6.1 lbs c) 28.5 lbs d) 33.6 lbs
18) The electrical resistance of a wire varies inversely as the
square of its diameter and directly as its length. If 225
meters of wire with a diameter of 0.3 cm has a resistance of
2.2 ohms, what is the resistance of 450 meters of wire
whose diameter is 0.7 cm?
a) 0.8 b) 2.9 c) 6.6 d) 8.1
19) Seven tractors can plow a field in 3 hours. How long
will it take five tractors?
a)2.1 hours b)11.7 hours c)4.2 hours d) 10 hours
20) Sally made 100$ when she worked 12 hours. How
many hours did she work if she made $150
a) 18 hrs. b) 8 hrs c) 1250 hrs. d) 16 hrs.
21) Simplify
a) 2x4y2
3
3
4x 2 y *
xy b)
3
3
x7 y 4
4 x 9 y 5 c)
3
4 x 4 y 2 d) x 3 y 3 4 y 2