Unit 1, Outcome 2 - Functions and Graphs

1. Sets


1. List all the numbers in the set P = { x ∈ ℕ : 1 < x < 5 }.

P =  { 2, 34 }

2. Functions


1. A function g is defined by g (x) = x - ( 6 / (x+4) ). Define a suitable domain for g.

 { x ∈ R : x ≠ 4 }.

2. A function f is defined by f (x) = sinx° for  x ∈ R. Identify its range.

y = sinx°

range = -1 < x < 1

3. Composite Functions


1. Functions f and g are defined by f(x) = 2x and g(x) = x - 3. Find:
a.) f(2)

f(2) = 2x
       = 2 x 2
        = 4

b.) f ( g(x) )

f ( g(x) ) = f ( x - 3)
              = 2(x - 3)
              = 2x - 6

c.) g ( f(x) )

g ( f(x) ) = g ( 2x )
             = 2x - 3

2. Functions f and g are defined on suitable domains by f(x) = x³ + 1 and g(x) = 1/x. Find formulae for h(x) = f ( g(x) ) and k(x) = g ( f(x) ).

h(x) = f ( 1/x )
       = (1/x)³ + 1

k(x) = g ( x³ + 1)
       = 1 / ( x³ + 1 )

4. Inverse Functions


1. f(x) = 5x so g(x) = (1/5)x

2. h(x) = 1/2 (1 - 5x) so k(x) = (1- 2x) / 5

5. Exponential Functions

6. Introduction to Logarithms


1. Sketch the curve with equation y = log6x

x = 1, y = log6(1) = 0    (1,0)
x = 6, y = log6(6) = 1    (6,1)

7. Radians


1. Convert from degrees to radians.
a.) 50° = 50 x (π/180) = (5/18)π radians
b.) 60° = 60 x (π/180) = π/3 radians

2. Convert from radians to degrees.
a.) π/4 = 180/4 = 45°
b.) π/6 = 180/6 = 30°

8. Exact Values


1. Find sinx + cosx, when x = π/6 radians.

sinπ/6 + cosπ/6 = (1/2) + (√3/2)
                         = (1 + √3)/2

10. Graph Transformations


1. Sketch the graph of y = -f(x) - 2

Firstly reflect in the x-axis then move down by 2.

2. Sketch the graph of y = 5cos(2x°) when 0 ≤ x ≤ 360.

NOTE:


TRANSLATION
y = f(x) + a  THEN if it's positive move up, negative move down.
y = f(x + a) THEN if it's positive move left, negative move right.

REFLECTION
y = - f(x) THEN flip in the x-axis
y = f(-x) THEN flip in the y-axis.


SCALING
y = kf(x) THEN stretches/compresses vertically.
If k > 1 THEN stretch.
If 0 < k < 1 THEN compress
y = f(kx) THEN stretches/compresses horizontally.

If k > 1 THEN stretch.
If 0 < k < 1 THEN compress

Always reflect before you translate, but scale before you reflect.