Functions (topic 2)

Question 1

Functions

Answer 1

One variable depends on the other

Question 2

Domain

Answer 2

Is the set of all valid input values (x-values) for a function. For this course, domain is limited by the following illegal operations:

1. division by zero
2. roots of negatives (greater than or equal to zero)
3. logs of zero and negatives (must be positive)
4. Certain values of tan

Question 3

Range

Answer 3

is the corresponding set of output values (y-values) for function. The graph and / or knowledge of asymptotes, maxima, minima should help determine the range.

Question 4

Composite functions

Answer 4

(fog)(x) is a composite function. It means f following g or f(g(x)). Start evaluating from the inside. In general, order matters in a composite function.

Question 5

Inverse functions

Answer 5

inverse f(x) is NOT 1 / f(x)

Question 6

Slope and y-intercept form

Answer 6

y = mx + c

Question 7

Point-slope form

Answer 7

y - y1 = m(x - x1)

Question 8

Slope or gradient

Answer 8

m = Δy / Δx = (y2 - y1) / (x2 - x1)

Question 9

To find y-intercept

Answer 9

Sub 0 for x

Question 10

To find the x-intercept

Answer 10

Sub 0 for y

Question 11

Quadratic function

Answer 11

Highest power x^2

Question 12

Types of quadratic functions

Answer 12

1. Vertex form
2. Factored form
3. Trinomial / expanded form

Question 13

Vertex form

Answer 13

y = a(x - h)^2 + k

• the vertex is (h, k)
• the a-value determines the direction of opening and steepness

Question 14

Factored form

Answer 14

y = a(x - p)(x - q)

• the values of p and q are the x-intercepts
• the vertex and axis of symmetry lie halfway between the x-intercepts
• the a-value determines the direction of opening and steepness
• parabolas that do not have x-intercepts cannot be written in this form

Question 15

Trinomial / expanded form

Answer 15

y = ax^2 + bx + c

• c is the y-intercept
• the a-value determines the direction of opening and steepness
• the axis of symmetry is x = -b / (2a) (x-coordinate of vertex)
• can be converted to “vertex form” through completing the square, using the axis of symmetry, or using differentiation techniques

Question 16

The discriminant

Answer 16

Δ = b^2 - 4ac

In a quadratic equation, discriminant tells us how many real solutions exist. In a quadratic function, discriminant tells us how many x-intercepts exist.

Question 17

If Δ < 0

Answer 17

There are no x-intercepts or real solutions

Question 18

If Δ = 0

Answer 18

There is one (repeated) x-intercept or real solution

Question 19

If Δ > 0

Answer 19

There are two distinct x-intercepts or real solutions

Question 20

Finding the horizontal asymptote

Answer 20

imagine y-value as x gets large (think lim x->infinity)

Question 21

Finding the vertical asymptote

Answer 21

what x-value makes the bottom zero (domain restriction?)

Question 22

Function transformations

Answer 22

• f (x - c): translate (shift) right c units
• f (x) + d: translate (shift) up d units
• pf(x): stretch vertically by a scale factor of |p|
  (note: negative p values cause a reflection through the x-axis)
• f (qx): stretch horizontally by a scale factor of |1 / q|
  (note: negative q values cause a reflection through the y-axis)