Inequality Notation vs. Interval Notation Table
a) -15 < x < 4
Interval Notation: [-15, 4)
b) [-6, 0) U [2, ∞)
c) x < 3 or x > 8
Interval Notation: (-∞, 3) U (8, ∞)
d) -7 < x ≤ -1
Interval Notation: (-7, -1]
For each relation, write letters of applicable descriptions:
a) f(x) = (x + 4)² + 2
Applicable Descriptions: A.
b) f(x) = 13|x| + 2
Applicable Descriptions: A, B, C.
c) [relation not specified]
Applicable Descriptions: A, C, D.
d) [relation not specified]
Applicable Descriptions: A, C, D.
e) [relation not specified]
Applicable Descriptions: none.
f) [relation not specified]
Applicable Descriptions: A, C, D.
g) [relation not specified]
Applicable Descriptions: A, C, D.
Describe end behavior:
a) As x → -∞, f(x) → ? (specific behavior not provided in text)
b) As x → +∞, f(x) → ? (specific behavior not provided in text)
Use graph to find limits:
a) lim f(x) as x → 3- = 7
b) lim f(x) as x → 3+ = 5
c) lim f(x) as x → -5- = 2
d) lim f(x) as x → -5+ = -10
e) lim f(x) as x → -1- = 3
f) lim f(x) as x → -1+ = -6
g) f(-1) = ? (value not provided)
h) Identify discontinuities:
Type at x = -5: jump discontinuity.
Type at x = -4: removable discontinuity.
Type at x = -1: removable discontinuity.
Type at x = 3: infinite discontinuity.
For each function:
a)
i) Domain: All real numbers except -4 ≤ x < -2
Interval: (-∞, -4) U [-2, ∞)
ii) Range: Real numbers ≥ -6
Interval: [-6, ∞)
iii) Behavior:
Increasing: -2 < x < 1
Interval: (-2, 1)
Decreasing: x < -4 and 1 < x < 4
Intervals: (-∞, -4) and (1, 4)
Constant: x = 4
Interval: (4, ∞)
b)
i) Domain: All real numbers
Interval: (-∞, ∞)
ii) Range: All real numbers except 2
Interval: (-∞, 2] U (4, ∞)
iii) Behavior:
Increasing: All real numbers
Interval: (-∞, ∞)
Decreasing: never
Constant: never
For given functions, find zero(s) and domain:
a) f(x) = 6x³ + 7x² - 3x
i) Zeros: (0, 0), (1, 0), (2, 0)
b) h(x) = 3x + 18
i) Domain: All real numbers
Interval: (-∞, ∞)
c) h(x) = √(x² - 10)
i) Domain: x < -√10 or x > √10
Accurately graph the following:
a) f(x) = √(x) + a
b) f(x) = 1/x + b
c) f(x) = [function not specified] ...
Sketch graphs based on transformations:
a) -1/(f)
b) f(-x) - 4
c) |f(x)|
d) f(2x) + 3
e) -2f(x - 2) + 1
f) |f(x)|
Graph the following piecewise functions:
a) f(x) = {
x - 3 if x < -1
3(x + 2) - 2 if -3 ≤ x < 0
[additional functions not specified]
b) f(x) = {
{function expressions provided}
[additional functions not specified]
Match operations to simplified forms:
(f ° g)(x) = D (f(g))
(f + g)(x) = B
(f - g)(x) = A
(fg)(x) = E
For given functions, find results:
a) (f + g)(x) = -2x² + x - 8
b) (g - f)(x) = 2x² + x - 14
c) (f ° g)(x) = -2x³ + 22x² + 3x - 33
Sketch inverse functions based on function graphs shown:
a) [original function not specified]
b) [original function not specified]
c) [original function not specified]
Write equations of inverse functions:
a) f(x) = √(3x - 12)
b) [original function not specified]
Find two functions such that h(x) = (f ° g)(x):
a) h(x) = (2x + 1)²
b) h(x) = √(9x - 14)