Math Analysis Ch. 1 Test review
What’s on the test:
1.2:
Finding Domain: (remember these are the x values)
*if there is not a radical or a fraction, your domain will be (-∞,∞)
**if there is a radical set it = 0 and solve, then decide what can and can’t be in the domain
**if there is a fraction, set the denominator = 0, then decide what can and can’t be in the domain
Ex: f(x) =
16 – x2 = 0
16 = x2
x = ±4
Domain: [-4, 4]
ex: x + 5
x–8
x–8=0
x = 8 (cannot be in domain
because 0 can’t be in
denominator)
try a #
< -4
bet. -4 & 4
>4
Not true, so not in
true so
not true, so not
Domain
[-4, 4]
in domain
***Remember -4 & 4 are included because you can have
a 0 under the radical
try a # < 8
4+5
4–8
9
-4
Denominator ≠0
So in domain
(-∞, 8)
>8
10 + 5
10 - 8
15
2
Denominator ≠0
so in domain
(8, ∞)
Domain: (-∞, 8) U (8, ∞)
Finding range: (remember these are the y values)
Graph the function on a calculator and look at the graph
Examples:
Range: [2, ∞)
r: (-∞, ∞)
R: (-∞, -2) U (0, ∞)
Local Exterma: (maximum & minimum)
*first determine how the graph is bounded: bounded above, bounded below, or bonded, then find the maximum and/or
minimum points. If the function is bounded above it only has a maximum, if it bounded below it only has a minimum, if it’s
bounded it has at least 1 maximum & 1 minimum
Examples: graph on the calculator, 2nd trace, then 3 or 4 (depending on which you are finding), follow the onscreen
instructions
1.f(x) = -2x2 + 3
has a maximum, so do 2nd trace, #4, left bound (to the left of the vertex) hit
enter, right bound (to the right of the vertex) hit enter, hit enter, maximum
point will show at the bottom of the screen: x= -1.938E-6, y = 3, so my answer
is : bounded above, maximum at (0, 3)
**remember your #’s may be a little different depending on where you put
your bounds)
2. f(x) = x3 – 9x
bounded, maximum: (-1.73, 10.39), minimum: (1.73, -10.39)
Increasing/decreasing intervals:
*based only on the domain, remember if there are arrows on the graph it continues to -∞ and/or ∞, just read the graph
Examples:
Decreasing:[-6, 0]
Decreasing: (∞, 1]
Increasing: [0, 6]
Increasing: [1, ∞)
Continuity/Discontinuity:
*3 types of discontinuity: infinite (when there is a VA), removable (when a point is removed from the graph - won’t seen a
hole on the calculator), and jump
*graph on the calculator: remember if it looks continuous, you must look at the table of values to see if the word “error” is
there, if you see “error” then it is removable discontinuity
Examples:
Continuous
removable discontinuity
jump discontinuity
infinite discontinuity
Odd/Even/Neither functions:
*odd functions: for all x in the domain of f, f(-x) = -f(x), symmetrical over the origin
* even functions: for all x in the domain of f, f(-x) = f(x), symmetrical over the y-axis
*neither: not symmetrical or doesn’t follow the above f(-x) = -f(x) and f(-x) = f(x)
Examples: even:
f(x) = x2
Odd:
f(x) = x3
neither:
f(x) = x3 – x + 1
Vertical Asymptotes/Horizontal Asymptotes:
*VA’s are vertical lines that the graph will not cross, always written as x = #
*HA’s are horizontal lines that the graph will not cross, always written as y = #
*to find VA’s: set the denominator = 0 but first factor both numerator & denominator & cancel (only if you can)
Examples:
4x
x2 – 16 = 0
x2 = 16
x = ±4: VA’s
2
x - 16
x+5 = x+5
, since the (x + 5)’s cancelled there is a hole at x = -5 so it’s not a VA, so
x2 – 25 (x – 5)(x + 5) x – 5= 0
x = 5 is the only VA
to find HA’s: you must look at the highest exponents in the numerator and denominator
if it’s higher in the numerator there is No HA
if it’s higher in the denominator the HA is y = 0
if they are equal, the HA is y = a/b (a and b are the leading coefficients)
examples:
1. 5x3
2. 6x – 1
3. 8x2 – 4x
3
2x + 1
x –1
2x2 – 4
3
3
2
x > x so
x < x so
x = x2 so y = 8/2, so
HA: none
HA: y = 0
HA: y = 0
1.3 The 12 basic functions:
*know what the 12 basic functions look like (either use the green paper I gave you or the last page in your textbook) and
be prepared to answer any questions about them (mainly what I assigned to you from 1.3)
1.4 building new functions:
*operations w/ functions, composition of functions, and evaluating functions, also finding domains
Examples: f(x) = x + 5, g(x) = 2x2 – 4
f(x) + g(x) = (x + 5) + (2x2 – 4) = 2x2 + x + 1, domain: (-∞, ∞)
f(x) – g(x) = (x + 5) – (2x2 – 4) = -2x2 + x + 9, domain: (-∞, ∞)
f(x)g(x) = (x + 5)(2x2 – 4) = 2x3 + 10x2 – 4x – 20, domain: (-∞, ∞)
f(x)/g(x) = (x + 5)/(2x2 – 4) = (x + 5)/2(x2 – 2), domain: (-∞, U ()U(
f(g(x)) = (2x2 – 4) + 5 = 2x2 + 1, domain: (-∞, ∞)
g(f(x)) = 2(x + 5)2 – 4 = 2(x2 + 10x + 25) – 4 = 2x2 + 20x + 50 – 4 = 2x2 + 20x + 46, domain: (-∞, ∞)
**remember (x + 5)2 ≠ x2 + 25 but is = x2 + 10x + 25**
Evaluate functions: f(x) = 2x + 5, g(x) = x – 2
(f ˚ g)(4) = 2(x – 2) + 5 = 2x -4 + 5 = 2x + 1 = 2(4) + 1 = 9
(g˚ f)(-2) = 2x + 5 – 2 = 2x + 3 = 2(-2) + 3 = -1
**remember if you get –x2 the “-“ is separate from the x2 and should be thought of as -1x2**
1.5 Parametric and inverse functions:
*Parametric functions: complete the table of values for the given values of t, graph the parametric function using a calculator
– DO NOT FORGET TO PUT THE CALCULATOR IN PARAMETRIC MODE, and find the direct algebraic
relationship
Example: find the direct algebraic relationship of x = 4t – 2, y = 2t
x = 4t – 2 (solve for t)
y = 2t
x + 2 = 4t
y=2 x+2
x+2 =t
4
4
y=x+2
2
*find inverses
1) Replace f(x) with y, 2) switch x and y values, 3) solve for y, 4) rewrite as f-1(x), 5) determine if the inverse if a function
Examples: 1. f(x) = x2 – 16
2. f(x) = x + 2
y = x2 – 16
x
x = y2 – 16
y=x+2
xy – 2 = y
x + 16 = y2
x
-2 = y – xy
=y
x=y+2
-2 = y(1 – x)
Yes
y
-2 = y
xy = y + 2
1-x
yes
*Verifying inverses: find f(g(x)) and g(f(x)), if f(g(x)) = g(f(x)) they are inverses
Example: f(x) = 3x - 2, g(x) = x + 2
3
f(g(x)) = 3 x + 2 - 2 = x +2 – 2 = x
3
g(f(x)) = 3x – 2 + 2 = 3x = x
3
3
Yes they are inverses since f(g(x)) = f(g(x))
1.6 transformations:
- identify the transformation that is taking place and graph both functions on the same graph
- Be prepared to transform any of the 12 basic functions
*Horizontal translations:
y = f(x – c): a translation to the right c units
y = f(x + c): a translation to the left c units
*Vertical translations:
y = f(x) + c: a translation up by c units
y = f(x) – c: a translation down by c units
*Reflections
y = - f(x): reflection across the x-axis
y = f(- x): reflection across the y-axis
y = - f(- x): reflection across the origin
*Stretches and Shrinks
Horizontal stretches/shrinks
y = f(x/c) : a stretch by a factor of c if c > 1
a shrink by a factor of c if c < 1
vertical stretches/shrinks
y = cf(x) : a stretch by a factor of c if c > 1
a shrink by a factor of c if c < 1