Download PPT Chapter 1 Review

Chapter 1 Review
Pre-Calc
Example
If f(x) = x2 + 3, evaluate f(2)
f( x2) = x2 2 + 3
f(2) = 4 + 3
f(2) = 7
What does this mean?
•It means when x = 2, y = 7
•It means the point (2,7) is on the graph
Example
If f(x) = 2x3 + 4x - 6, evaluate f(-1)
3+4
f( -1
x ) = 2 (-1)
(-1)
x- 6
x
f(-1) = 2(-1) + 4(-1) - 6
f(-1) = -12
What does this mean?
•It means when x = -1, y = -12
•It means the point (-1,-12) is on the
graph of f(x)
Example
If g(x) = x2 + 2x, evaluate g(x – 3)
g(x) = (x -3)
x2 + 2 (xx-3)
g(x) = (x2 – 6x + 9) + 2x - 6
g(x) = x2 – 6x + 9 + 2x - 6
g(x) = x2 – 4x + 3
Adding and Subtracting
Functions
Let f x   3x  8 and gx   2 x  12.
Find f  g and f - g
 f  g ( x)  f x   g x 
 f  g ( x)  f x   g x 
 (3x  8)  (2 x  12)
 5x  4
 (3x  8)  (2 x  12)
 x  20
Multiplying Functions
Let f x   x - 1 and gx   x  1.
Find f  g
2
f x   g ( x)  ( x  1)( x  1)
2
 x3  x 2  x  1
Dividing Functions
Let f ( x ) = 3x - 5 and g ( x ) = x - 9.
2
æf ö
Find ç ÷
ègø
f ( x ) 3x - 5
= 2
g ( x) x - 9
What values of x are not in the domain?
x -9 ¹0
2
x2 ¹ 9
x ¹ ±3
Let’s Try Some
Let f x   5x 2 -1 and gx   5x  1.
Find f x  g ( x)
What is the domain?
Find f x  g ( x)
Let’s Try Some
Let f x   5x 2 -1 and gx   5x  1.
Find f x  g ( x)
What is the domain?
Find f x  g ( x)
Let’s Try Some
Let f x   6 x 2  7x - 5 and gx   2 x  1.
Find f x g ( x)
What is the domain?
Find
f x 
g(x)
Let’s Try Some
Let f x   6 x 2  7x - 5 and gx   2 x  1.
Find f x g ( x)
What is the domain?
Find
f x 
g(x)
Example – Composition of
Functions
Let f ( x) = x - 2 and g ( x ) = x . Find ( g f ) ( x )
2
g  f x  g f x
= (x - 2)
2
x - 4x + 4
2
Let’s try some
Let f x   x3 and g x   x 2  7. Find g  f 2
( g f ) ( 2) = g ( f ( 2))
= g(8)
= 71
Solution
Let f x   x3 and g x   x 2  7. Find g  f 2
Find the zero of the function: f (x) = 2x + 5
2x + 5 = 0
2x = -5
-5
x=
2
FINDING INVERSE FUNCTIONS
2
f
(x)

x
Find the inverse of
STEPS
COMPUTATIONS
 with y
Replace f (x)
yx
2
xy
2
Interchange the roles of x
and y
Solve for y
x 

y
Replace y with f -1(x)


y
2
x
f -1 (x) = x
FINDING INVERSE FUNCTIONS
Find the inverse of f (x) = 4x + 5
STEPS
COMPUTATIONS
Replace f (x) with y
y  4x  5
Interchange the roles of x
and y
x  4y  5
Solve for y
x  5  4y

x 5
y
4
Replace y with f -1(x)

f 1 (x) 

x 5
4
Find the inverse of f (x) = 2x3 - 1
STEPS
COMPUTATIONS
y  2x 1
3
Replace f (x) with y
Interchange the roles of x
and y
x  2y 1
3
x  1  2y 3
Solve for y
x 1
 y3
2


y
Replace y with f -1(x)

f
1
3 x 1
2
(x) 
3
x 1
2
Find the inverse of
STEPS
Replace f (x) with y
Interchange the roles of x
and y
Solve for y
f (x) = x -1
COMPUTATIONS
y = x -1
x=
y -1
( x ) = ( y -1)
2
2
x 2 = y -1
y = x 2 +1
Replace y with f -1(x)
f
-1
(x) = x 2 +1
Find the distance between (5, 0) and
(0, 5)
d = (x2 - x1 ) + (y2 - y1 )
2
d = (0 - 5) + (5- 0)
2
d = (-5) + (5)
2
2
2
2
d = 25+ 25 = 50 = 5 2
Find the distance between (-1, 3)
and (2, 4)
d = (x2 - x1 ) + (y2 - y1 )
2
d = (2 - (-1)) + (4 - 3)
2
d = (3) + (1)
2
2
2
2
d = 9 +1 = 10
Find the distance between (0, 2b)
and (a, b)
d = (x2 - x1 ) + (y2 - y1 )
2
d = (a - 0) + (b - 2b)
2
2
d = (a) + (-b)
2
d = a +b
2
2
2
2
Find the midpoint of (10, 5) and (-1, -2)
æ x1 + x2 y1 + y2 ö
M (x, y) = ç
,
÷
è 2
2 ø
æ 10 + (-1) 5 + (-2) ö
M =ç
,
÷
è
2
2 ø
æ9 3ö
M =ç , ÷
è2 2ø
Various Forms of an Equation
of a Line.
Slope-Intercept
Form
Standard
Form
y  mx  b
m  slope of the line
b  y  intercept
Ax  By  C
A, B, and C are integers
A  0, A must be postive
y  y1  m  x  x1 
Point-Slope
Form
m  slope of the line
 x1 , y1  is any point
Write the equation of a line in slope-intercept form
that passes through points (3, -4) and (-1, 4).
4--4
y2 – y1
=
m=
x2 – x1
-1-3
=
y2 – y1 = m(x – x1)
Use point-slope form.
y + 4 = – 2(x – 3)
Substitute for m, x1, and y1.
y + 4 = – 2x + 6
Distributive property
y = – 2x + 2
8
–4
= –2
Write in slope-intercept form.
4.
Write an equation of the line that passes through
(–1, 6) and has a slope of 4.
ANSWER
5.
y = 4x + 10
Write an equation of the line that passes through
(4, –2) and is parallel to the line y = 3x – 1.
ANSWER
y = 3x – 14
Write equation of the line in standard form that has a
slope of ½ and passes through (4,-5).
y + 5 = ½(x – 4)
y + 5 = ½x - 2
y = ½x - 7
2y = x - 14
-x
-x
-x + 2y = -14
x - 2y – 14 = 0
Multiply everything by 2 to get rid
of the fraction
Write equation of the line in standard form that is
parallel to y=⅔x-8 and passes through (6,4)
m=⅔
y – 4 = ⅔(x – 6)
y – 4 = ⅔x - 4
y = ⅔x
3y = 2x
-2x
-2x
-2x + 3y = 0
2x - 3y = 0
Multiply everything by 3 to get rid
of the fraction
Write an equation in standard form of the line that
passes through (5, 4) and has a slope of –3.
SOLUTION
y – y1 = m(x – x1)
Use point-slope form.
y – 4 = –3(x – 5)
Substitute for m, x1, and y1.
y – 4 = –3x + 15
Distributive property
y = –3x + 19
+3x
+3x
3x + y = 19
Write in slope-intercept form.
Write equations of parallel or perpendicular lines
b. A line perpendicular to a line with slope m1 = –4 has
a slope of m2 = – 1 = 1 . Use point-slope form with
m1 4
(x1, y1) = (–2, 3)
y – y1 = m2(x – x1)
1
(x – (–2))
4
1
y–3=
(x +2)
4
1
1
y–3= x+
4
2
y–3=
1
7
y  x
4
2
Use point-slope form.
Substitute for m2, x1, and y1.
Simplify.
Distributive property
Write in slope-intercept form.
Michael Jordan is producing a new
line of shoes. If he were to produce
500 shoes he would profit $20,000. If
he were to produce 150 shoes it would
cost him $6,000. Write a linear
equation relating cost, y, to the
number of shoes, x.
( 500,
(150,
20000)
6000)
20000 - 6000
m=
= 40
500 -150
y - 6000 = 40 ( x -150)
y - 6000 = 40x - 6000
y = 40x