01 Graphs

Question 1

(fg)^-1

Answer 1

(fg)^{-1 =} f^-1g^-1

Question 2

f f-1 (x)

Answer 2

f f-1 (x)=x

Question 3

Domain D

Answer 3

the range given in the qn
Usually x axis

Question 4

Range

Answer 4

Extent of Y axis

Question 5

() vs []

Answer 5

() not inclusive
[] including that value

Question 6

how to get f^-1 from f(x)

Answer 6

make x the subject
replace x with f^-1

Question 7

Test for Existence of Functions

Answer 7

only if any vertical line x = k, where k is a constant, k ∈ Df
cuts the graph at one and only one point.

Question 8

Test for inverse functions

Answer 8

to check if it is one-to-one
use the horizontal line test
f is one-one if every horizontal line y = k, k ∈ R
cuts the graph at one point.

Question 9

Relationship between a Functions and its Inverse

Answer 9

The point(s) of intersection of g and g<sup>-1 </sup>lie on the line y = x
Function and its Inverse are reflected images of each other in the line y = x

Question 10

Inverse functions range/domain

Answer 10

R<sub>f-1</sub> = D<sub>f</sub>
R<sub>f</sub> = D<sub>f-1</sub>

Question 11

Test for composite function

Answer 11

R₁ must be a subset to D₂

Question 12

Domain of composite function gf

Answer 12

Dgf = Df

Question 13

Range of composite function gf

Answer 13

take Df domain of first
insert into domain of second Dg
Find Rg range of second
Yea thats the range of composite Rgf woo

Question 14

f f-1 (x) = f-1 f(x) = x
Inverse composite domain

Answer 14

The domain of f-1 f(x) is Df
The domain of ff-1 (x) is Df -1
Domain follows the first function

Question 15

Finding asymptotes

Answer 15

Oblique: Long division if improper frac
Vertical: Let denom be 0
divide coeff of numerator and denominator

Question 16

Eqn of circles

Answer 16

(𝑥 − ℎ) 2+(𝑦 − 𝑘) 2= 𝑟 2 , where 𝑟 ≠ 0

Question 17

Eqn of ellipses

Answer 17

(𝑥 − ℎ)²/𝑎² + (𝑦 − 𝑘)²/𝑏² = 1

Question 18

Eqn of hyperbola and asymptotes

Answer 18

(𝑥 − ℎ)²/𝑎² - (𝑦 − 𝑘)²/𝑏² = 1
𝑦 = ± 𝑏/𝑎 (x-h) + k

Question 19

Scaling

Answer 19

replace by y/2 to x2
replace by 2y to halve

Question 20

Reflection

Answer 20

-y = f(x) flips x axis
y = f(-x) flips y axis

Question 21

Translation

Answer 21

y-1 shift by 1 unit positive y direction
y+1 shift 1 unit neg y direction

Question 22

y = F(|x|)

Answer 22

Keep all right hand side +x and copy over

Question 23

drawing f(x) to f’(x)

Answer 23

horizontal asymptotes become y=0
vertical asymptotes remain

Question 24

f(x) to 1/f(x) drawing

Answer 24

asymptotes become x intercepts
x intercepts become asymptotes
max pt to min pt
min pt to max pt
oblique symptotes to horizonal y=0