Chap 2 - functions

Question 1

types of functions

Answer 1

1) 1 to 1 - 1 x value has only 1 y value eg. linear eq
2) many to 1 - 1 x value has only 1 y value, but 1 y value can have many x values eg, quadratic
3) 1 to many - 1 x value has more than 1 y value eg. eq with √

Question 2

domain & range

Answer 2

domain - x value
range - y value
1) domain & range without limits would both be ∈ ℝ (all real no.)
2) if there is an asymptote, domain & range both need 2 limits
3) if x has an even power in eq, range is limited, else it is ∈ ℝ
4) for quadratics, draw rough shape of eq, TP shows range limits & smallest number x can be for the graph to be 1 to 1

Question 3

composite functions

Answer 3

-order matters
- f^-1 f (x) = ff^-1 (x) = x

Question 3

joining different functions tgt

Answer 3

-draw eq until the limit
-empty circle means >/<
-coloured in circle means ≥/≤

Question 4

finding inverse function

Answer 4

-inverse function only exists if function is 1 to 1
-swap x & y values & solve
-domain of inverse = range of function
-range of inverse = domain of function
-so TP of inverse is x & y values of TP of function but swapped

Question 5

how an inverse quadratic function looks

Answer 5

-flip eq to the side so that half is in + & - quadrants
-y = x is line of symmetry
-if inverse eq = original function, eq is self inverse because both graphs are the same

Question 6

reflections

Answer 6

1) reflect at x axis = -(x ) (x ) / -f (x)
-x intercept is constant
2) reflect at y axis = (-x ) (-x ) / f (-x)
-y intercept is constant
- b signs are opposite of that in original eq

Question 7

translations

Answer 7

1) usually complete the square
2) (0 a) moves along y axis - use the same number in eq
y = f(x) + a
(a 0) moves along x axis - use the opposite sign for the num. in eq
y = f(x-a)
for inverse eq:
-move along y axis - a √x
-move along x axis - √ (x + a) - a is opposite sign to the one in the given translation

Question 8

stretches

Answer 8

1) // to y axis = x intercept is constant
-if a >1, vertical stretch
-if a <1, horizontal stretch
a f(x)
2) // to x axis = y intercept is constant
-if factor is a, horizontal stretch
-if factor is 1/a, vertical stretch
by a factor of 1/ a - take inverse of a
f(ax) , (just take inverse of factor & put the power on x to it) multiply by coefficient in front of x

Question 9

sequence of transformations

Answer 9

V = vertical, H = horizontal
1) V + H / H + V does not affect outcome
2) V + V has to be stretch, translate
3) H + H has to be translate + stretch