Combined Study Sheet
Unit 1, Part 1:
- Inverses are reflected across the line y = x
- To convert absolute value into piecewise;
○ Find when abs. value function's zeroes
○ Find when its positive / negative
○ Split function into intervals, negate the interval that the function is negative
-
- Inverse of a function only exists if the original function passes the Horizontal Line Test. To ensure
this happens, limit the domain of the original function to be 1-to-1
Unit 1, Part 2:
-
- Given |x|< a, solution is;
○ -a < x < a
- Given |x| > a, solution is
○ x < -a, x > a
- The combination of functions must include the overlap of the domains of each function
-
- Revenue, R(x), is Profit, P(x) * # of buyers, N(x)
(f ∘ g)(x) = (f(g(x))
- (𝑓 ∘ 𝑔)(𝑥) ≠ (𝑔 ∘ 𝑓)(𝑥) ,
- Inner function's range and its overlap with outer function's domain
- If g(x) is the inverse of f(x),
𝑓(𝑔(𝑥) = 𝑔𝑓(𝑥)= 𝑥
- Steps to finding domain of composite function;
○ Find domain of every function
○ Domain of composite = overlap each of function
( )
Reciprocal =
( )
, Rational =
( )
( )
Reciprocal Characteristics:
V.As at g(x) at Zeroes of f(x)
HA at y = 0
Interval of Increase on f(x) =
Interval of Decrease on g(x)
Local Max at (a, f(a)) = Local Min at
(a,
)
( )
Interval of Decrease on f(x) = Local Min at (a, f(a)) = Local Max at
Interval of Increase on g(x)
(a,
)
( )
Same Positive / negative
intervals as f(x)
- To find behaviour of function near asymptotes, use interval chart and then limit notation
- To find oblique asymptote, use long division, where numerator = inside, denominator =
divisor, quotient = equation of asymptote
- When solving rational equations, only the numerator matters
- When solving rational inequalities, use an interval chart
Unit 3
Radian (𝜽): angle at center of circle whose arc length = radius length
Radius (𝒓): straight length from circle's center to circumference
Arc Length: 𝑎 = 𝜃𝑟
Revolutions * 2𝜋 = distance in radians
𝑟𝑎𝑑
𝜋
=
𝑑𝑒𝑔 180
𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑟𝑎𝑑𝑖𝑎𝑛𝑠)
𝑇𝑖𝑚𝑒 (𝑠𝑒𝑐𝑜𝑛𝑑𝑠)