0 - Review of Fundamentals

Question 1

A function f is linear if it obeys the equation:

f(ax₁ + bx₂) = _________

Answer 1

A function f is linear if it obeys the equation:

f(ax₁ + bx₂​) = af(x₁) + bf(x_{<span>2</span>})

Question 2

Probing a linear system with each ____ ______ is enough to completely characterize the system.

Answer 2

Probing a linear system with each input direction is enough to completely characterize the system.

Question 3

In function notation we have the “zeros of f” whereas analogously in matrix notation we have the “_____ of T”

Answer 3

In function notation we have the “zeros of f” whereas analogously in matrix notation we have the “kernel of T”

Question 4

Invertible transformations are one-to-one correpondences or __jections between vectors in the input space and vectors in the output space.

Answer 4

Invertible transformations are one-to-one correpondences or bijections between vectors in the input space and vectors in the output space.

Question 5

Given f(x) = ax²+bx+c = (x-x₁)(x-x₂), we know that the two numbers x₁ and x₂ are called the roots of the function and are the points where the function f(x) touches the _-____.

Answer 5

Given f(x) = ax²+bx+c = (x-x₁)(x-x₂), we know that the two numbers x₁ and x₂ are called the roots of the function and are the points where the function f(x) touches the x-axis.

Question 6

There is a quick test to check if a quadratic function has roots (or touches or cross the x-axis).

If b²-4ac > 0, Then _____

If b²-4ac = 0, Then _____

If b²-4ac < 0, Then _____

Answer 6

There is a quick test to check if a quadratic function has roots (or touches or cross the x-axis).

If b²-4ac > 0, Then f(x) has 2 roots

If b²-4ac = 0, Then f(x) has only 1 root

If b²-4ac < 0, Then f(x) has no roots or doesn’t touch x-axis

Question 7

The ______ of a function is the set of allowed inputs.

The _____ or _____ of a function is the set of all possible output values.

The ______ of a function describes the type of the outputs the function has.

Answer 7

The domain of a function is the set of allowed inputs.

The image or range of a function is the set of all possible output values.

The codomain of a function describes the type ​of the outputs the function has (like integers, real numbers, etc).

Question 8

A function is ________ or ______ if it maps different inputs to different outputs

(think of injecting a fluid or shot into a cylindrical pipe or arm and watching the fluid spread out with no two inputs going to the same output).

Answer 8

A function is one-to-one or injective if it maps different inputs to different outputs

(think of injecting a fluid or shot into a cylindrical pipe or arm and watching the fluid spread out with no two inputs going to the same output).

Question 9

A function is ____ or _______ if it covers the entire output set.

In other words, the image of the function is ____ to the function’s codomain.

(think of surveying the entirety of the land)

Answer 9

A function is onto or surjective if it covers the entire output set.

In other words, the image of the function is equal to the function’s codomain.

(think of surveying the entirety of the land)

Question 10

A function is ______ if it is both _____ and _____.

In this case, it could also be said that f is in __-__-__ _______ between the input set and output set.

In other words, for every possible output there exists exactly one input.

Answer 10

A function is bijective if it is both injective and surjective.

In this case, it could also be said that f is in one-to-one correspondence between the input set and output set.

In other words, for every possible output there exists exactly one input.

Question 11

Given a bijective function f, there exists an ____ function f^-1 that performs the ____ mapping of f.

In other words,

f^-1(f(x)) = f^-1 . f (x) = ____

Answer 11

Given a bijective function f, there exists an inverse function f^-1 that performs the inverse or reverse mapping of f.

In other words,

f^-1(f(x)) = f^-1 . f (x) = x

Question 12

The equation of a line describes input-output relationships where the change in the output is ________ to the change in the input.

Answer 12

The equation of a line describes input-output relationships where the change in the output is proportional to the change in the input.

Question 13

The equation of the line can be reformulated in a more symmetric form as the general equation and as a relation:

Ax + __ = __

where the traditional m = ___, and b = ____

Answer 13

The equation of the line can be reformulated in a more symmetric form as the general equation and as a relation:

Ax + Bx = C

where the traditional m = -A/B, and b = C/B

Question 14

What is the inverse function of the function f(x)=x²?

What graph does this start with and what does this inverse function transform it into? (How do they look when graphed?)

Answer 14

What is the inverse function of the function f(x)=x²?

f(x) = sqrt(x)

What graph does this start with and what does this inverse function transform it into? (How do they look when graphed?)

The y’s and x’s switch places. Thus, the former parabola that faces up is not turned sideways and opens to the right, with one side on the positive quadrant and one side below the x-axis in the negative quadrant. Notice how we now have to take out the negatives below y=0 to pass the vertical line test to still be considered a function.

Question 15

An even polynomial contains only ____ powers of x. It has the property that f(x) = ____.

An odd polynomial contains only ____ powers of x. It has the property that f(x) = ____.

If a polynomial has both even and odd terms then it is _____.

Answer 15

An even polynomial contains only even powers of x. It has the property that f(x) = f(-x).

An odd polynomial contains only odd powers of x. It has the property that f(x) = -f(x).

If a polynomial has both even and odd terms then it is neither even nor odd.

Question 16

Properties of sin(x) function:

sin(x) is an even/odd function

Domain:

Range / Image:

Roots:

Inverse function:

Answer 16

Properties of sin(x) function:

sin(x) is an odd function

Domain: Reals

Range / Image: y in [-1, 1]

Roots: all multiples of π (anytime it touches x-axis)

Inverse function: sin^-1(x) which is NOT the same as (sin(x))^-1=csc(x)

Question 17

Properties of the exponential function:

Domain:

Image:

Derivative:

Answer 17

Properties of the exponential function:

Domain: Reals

Image: (0, +inf)

Derivative: e^x

Question 18

We can summarize the polynomial notation of,

f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x¹ + a₀

…into something more succinct, like….

Answer 18

We can summarize the polynomial notation of,

f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x¹ + a₀

…into something more succinct, like….

the sum of products notation:

f(x) = _nΣ_k=0 (a_k•x^k)

Question 19

What’s an easy way to solve for x in the following equation?

g(x) = x⁴ - 7x² + 10 = 0

Answer 19

What’s an easy way to solve for x in the following equation?

g(x) = x⁴ - 7x²​ + 10 = 0

Substitution trick:

Substitute y=x²

Question 20

The trigonometric identity

cos²x + sin²x = 1

is really the ________ Theorem. Derive this.

Answer 20

The trigonometric identity

cos²x + sin²x = 1

is really the Pythagorean Theorem. Derived:

|adj|² + |opp|² = |hyp|²

|adj|²/|hyp|² + |opp|²/|hyp|² = 1

cos²x + sin²x = 1 ​

Question 21

sin(30) = sin(__/__) =____

Knowing this, use the fact that cos²x + sin²x = 1 to find cos(30)

Answer 21

sin(30) = sin(π/6) = 1/2

Knowing this, use the fact that cos²x + sin²x = 1 to find cos(30)

cos(30) = sqrt( 1 - sin²(30) )

cos(30) = sqrt( 1 - (1/2)²)

cos(30) = sqrt(3) / 2

Question 22

Generally for non-unit circles, cos(theta) = adj/hyp = r_x/r,

so r_x = _____

Answer 22

Generally for non-unit circles, cos(theta) = adj/hyp = r_x/r,

so r_x = rcos(theta)

Question 23

sin( a + b ) = ______

(Hint: mnemonic for this identity is sico + sico)

Answer 23

sin( a + b ) = sin(a)cos(b) + sin(b)cos(a)

(Hint: mnemonic for this identity is sico + sico)

Question 24

cos( a + b ) = ______

(Hint: mnemonic for this identity is coco + sisi)

Answer 24

cos( a + b ) = cos(a)cos(b) + sin(a)sin(b)

(Hint: mnemonic for this identity is coco + sisi)

Question 25

The sin rule states

a / sin(alpha) = ______ = ______

Answer 25

The sin rule states

​a / sin(alpha) = b / sin(beta) = c / sin(gamma)

Question 26

The cos rule states that:

a² = ___ + ___ - _________

b² = ___ + ___ - _________

c² = ___ + ___ - _________

Answer 26

The cos rule states that:

a² = b² + c² - 2bccos(alpha)

b² = a² + c² - 2accos(beta)

c² = a² + b² - 2abcos(gamma)

Question 27

Circumference of a circle:

Answer 27

Circumference of a circle:

C = 2πr

Question 28

Area of a circle:

Answer 28

Area of a circle:

A = πr²

Question 29

Equation of a sphere:

Answer 29

Equation of a sphere:

x² + y² + z² = r²

Question 30

Surface area of a sphere:

Answer 30

Surface area of a sphere:

A = 4πr²

Question 31

Volume of a sphere:

Answer 31

Volume of a sphere:

V = 4/3 πr³

Question 32

Surface area of a cylinder:

Answer 32

Surface area of a cylinder:

A = 2(πr²) + (2πr)h

Question 33

Volume of a cylinder:

Answer 33

Volume of a cylinder:

V = (πr²)h

Question 34

Surface area of a cone with radius r and height h:

Answer 34

Surface area of a cone with radius r and height h:

A = πr² + πr sqrt( r² + h² )

Question 35

Volume of a cone:

Answer 35

Volume of a cone:

V = 1/3 π hr²

Question 36

The word vector comes from the latin vehere which means __ _____ (similar to vehicle)

Answer 36

The word vector comes from the latin vehere which means to carry (similar to vehicle​)

Question 37

The cross product of two vectors is only defined for ___-dimensional vectors.

Answer 37

The cross product of two vectors is only defined for 3-dimensional vectors.

Question 38

A coordinate system is the same as the _____, which is the set of vectors that can represent any vector in that system.

(for example, in colors the ____ is aRed + bGreen + Blue)

Answer 38

A coordinate system is the same as the basis, which is the set of vectors that can represent any vector in that system.

(for example, in colors the basis is aRed + bGreen + Blue)

Question 39

One such example of vectors is found when we do operations with the ____ number system.

Answer 39

One such example of vectors is found when we do operations with the complex number system.