TOPIC 5: FUNCTIONS

Question 1

Set Notation VS Interval Notation

Answer 1

Set Notation: {x ∈ ℝ: 3<x≤8}
Interval Notation: (3,8]

*Interval Notation only applies for real numbers

Question 2

Express y = 2x - 1 in different forms

Answer 2

f(x) = 2x - 1
f: x ↦ 2x - 1

Question 3

Express x = sint in different forms

Answer 3

f(t) = sint
f: t ↦ sint

Question 4

When does an equation represent a function?

Answer 4

When any vertical line intersects the graph EXACTLY ONCE (through vertical line test)

Question 5

Vertical Line Test phrasing

Answer 5

For any vertical line, x=k, where k∈ℝ, the line intersects the graph exactly once.

Question 6

Which is the input variable, rule and domain?

f(x) = 2x² + 1, x∈ℝ, x>0, x≠2

Answer 6

Input variable: x
Rule: 2x² + 1
Domain: (0,∞) \ {2}

Question 7

What is range?

Answer 7

The set of output values

Question 8

When are two functions equal to each other?

Answer 8

Same rule AND domain

Question 9

How to know if an inverse function exists?

Answer 9

If the function is a one-to-one function

Question 10

One-to-one function test phrasing

Answer 10

For any horizontal line, y=k, where k∈ℝ, the line intersects the graph exactly once only. So, f is one-to-one.

Question 11

Domain and range of inverse function, f⁻¹

Answer 11

Df-1 = Rf
Rf-1 = Df

Question 12

Steps to find f inverse

Answer 12

1. Make x the subject
2. Change y to x to get the rule
3. Find the range of f to get the domain of f inverse

Question 13

Features of graph of inverse function

Answer 13

1. f and f inverse are reflections of each other at y=x
2. Points (a,b) corresponds to points (b,a)

Question 14

To take note of when drawing inverse function graph

Answer 14

1. Equal scale for both x-axis and y-axis
2. Line y=x added as a dotted line, at angle of 45°
3. If f graph pointing away from y=x, inverse f must be the same as well

Question 15

When does composite function fg exist?

Answer 15

If R(g) ⊆ D(f)

Question 16

When does composite function gf exist?

Answer 16

When R(f) ⊆ D(g)

Question 17

Equation of fg

f: x ↦ 2x + 1 ; g: x ↦ x² - 1

Answer 17

2(x²-1) + 1 = 2x² - 1
fg: x ↦ 2x² - 1

Question 18

Domain of fg

Answer 18

Domain of g

Question 19

Range of fg

Answer 19

1. Sketch graph of fg to find range
2. Restrict domain of second function (f) to find range

Question 20

What are ff⁻¹ / f⁻¹f called?

Answer 20

Identity function

ff⁻¹(x) = x
f⁻¹f(x) = x