Functions

Question 1

What is the definition of a function?

Answer 1

a rule that assigns each element of an input to a unique output

Question 2

Answer 2

Question 3

Answer 3

Question 4

What is the domain convention?

Answer 4

If a function is given by a formula and the domain is not specified, the convention is that the domain is the set of all real numbers for which the formula defines a real number.

Question 5

Answer 5

Question 6

Answer 6

a is a function, b and c is not a function

Question 7

Answer 7

Question 8

What is a function of the degree 0

Answer 8

Just a constant real number

Question 9

What does the graph f(x)=x^1/n look like when n is even

Answer 9

Question 10

What does the graph f(x)=x^1/n look like when n is odd

Answer 10

Question 11

What does the graph f(x)=x^a look like when a is negative

Answer 11

also called the reciprocal function

Question 12

What is an absolute function?

Answer 12

An absolute value function is a function whose rule involves absolute value
bars.

An absolute value function can be written as a piecewise function.

Question 13

Answer 13

Question 14

What is a rational function?

Answer 14

Unless otherwise specified, the domain of f consists of all real numbers x such that q(x)≠ 0.

Question 15

Why does the rational function f(x)=p(x)/q(x) not have a value of q(x)=0?

Answer 15

There may a verticle assymptote at q(x)=0

Question 16

What is the difference between exponential function and power function

Answer 16

Exponential function has x as the index. Power function has any real number as the index.

Question 17

Define an exponential function

Answer 17

Question 18

Given a function a_x

What does the graph look like if a is below 1?

Answer 18

The graph ‘points’ left with a horizontal assymptote at x=0

Question 19

Given a function a_x

What does the graph look like if a is above 1?

Answer 19

The graph ‘points’ right with a horizontal assymptote at x=0

Question 20

True or false?

All functions with the form a^x where a is a real number pass through the point (0, 1)

Answer 20

True

Question 21

What is the formula for turining point of a prabola?

Answer 21

x=-b/2a

Question 22

Determine the domain and range of the following function then sketch it.

Answer 22

Question 23

Determine the domain and range of the following function then sketch it.

Answer 23

Question 24

Determine the domain and range of the following function then sketch it.

Answer 24

Question 25

# Determine the domain and range of the following function then sketch it.

Answer 25

Question 26

In a hyperbola (1/x). What does it look like when x is positive. (which parts of the cartesian plane is the line in)

Answer 26

The line is in the top right and bottom left. ## Footnote Think about the CAST circle. The top right is when all is **positive** and just think from there

Question 27

In a hyperbola (1/x). What does it look like when x is negative. (which parts of the cartesian plane is the line in)

Answer 27

The line is in the top left and bottom right. ## Footnote Think about the CAST circle. The top right is when all is **positive** and negative is just when its not there.

Question 28

Answer 28

Question 29

When a polynomial has an even degree (x⁴, x⁶, x⁸), does the polynomial look more like a quadratic or cubic?

Answer 29

Quadratic. It enters going down and exits going up

Question 30

When a polynomial has an odd degree (x⁵, x⁷, x⁹), does the polynomial look more like a quadratic or cubic?

Answer 30

Cubic. It enters going up and exits going up

Question 31

What are the three types of power functions?

Answer 31

1. Polynomial: where the index is a positive integer 2. Root function: where the index is a positive number between 0 and 1 (decimal index) 3. Reciprocal/hyperbola function: where the index is just -1 | Recall power functions have the form x^a for any real number a

Question 32

# Given a function x^1/a for some integer a It is known that the graph represent some sort of inverse of a polynomial to the degree a. **What is the domain of the 'inverse' function when a is even?**

Answer 32

[0, ∞)

Question 33

# Given a function x^1/a for some integer a It is known that the graph represent some sort of inverse of a polynomial to the degree a. **What is the domain of the 'inverse' function when a is even?**

Answer 33

(-∞, ∞)

Question 34

Determine the assymptote(s) of the following function

Answer 34

x=-1, 1

Question 35

Answer 35

Question 36

# Consider the function f(x)=x²-x-2 Determine the values of x for which x² − x − 2 ≥ 0.

Answer 36

Question 37

Answer 37

Question 38

Answer 38

Question 39

Write |7x+2| as a piecewise function (split at 0)

Answer 39

Question 40

What is meant by a composite function?

Answer 40

"A function applied to a function". It is when the the input of a function uses the output of another function. | f(g(x)) is called the composite function of f and g

Question 41

# Given a composite function f(g(x)) How would you denote that f(g(x)) is a composite function of f and g?

Answer 41

f ∘ g.

Question 42

# Given f ∘ g Which function do you apply first?

Answer 42

g | Just remeber f(g(x)) means f ∘ g and f is on the outside so its first.

Question 43

Answer 43

Question 44

What is the definition of a one-to-one function?

Answer 44

A function that where all values of the domain have only one unique image in the range. | all x's have only one y ## Footnote For proof: if f(x)=f(a) then x=a

Question 45

# Let f(x) be the function defined by f(x) = x² Determine a suitable domain for f so that f⁻¹ exists.

Answer 45

f(x) is not one to one so we must split it in two (at y=0), the two functions individually become one to one. So the domain is either (-∞, 0] or [0, ∞)

Question 46

Answer 46

Question 47

# True or false? The inverse of any function is obtained by reflecting it on the line y=x

Answer 47

True

Question 48

Answer 48

Question 49

Sketch the inverse of a sin function

Answer 49

## Footnote Initially, you cannot get an inverse of a sin as it is not one-to-one. But when you restrict the domain, it is possible which gives you the arcsin(x) function

Question 50

What is the domain and range of f(x)=arcsin(x)

Answer 50

Domain: [1, 1] Range: [-π/2, π/2]

Question 51

Sketch the inverse of a cosine function

Answer 51

## Footnote Initially, you cannot get an inverse of a cos as it is not one-to-one. But when you restrict the domain, it is possible which gives you the arccos(x) function