Mathematics: Definitions (ACADEC '25-'26)

If x and y are real numbers, what is the real number when their difference is x - y?

x + (-y)

If y ≠ 0, what is x/y as a real number?

x ⋅ y^-1

When does a polynomial K(x) divide a polynomial P(x)?

If there exists a polynomial Q(x) such that P(x) = K(x)Q(x)

What is a function ‘f’ a correspondence between?

Two sets, A and B, that assigns to each element x in A one and only one element f(x) in B

When does c = log_a(b)?

If and only if a^c = b

What operations does the set of real numbers denoted by R equipped with?

Addition and multiplication

What special property do the numbers 0 and 1 have?

1. For any real number x, x + 0 = x

2. For any real number x, x ⋅ 1 = x

What is the additive inverse property?

For any real number x there is a unique number, denoted by -x, such that x + (-x) = 0

What is the multiplicative inverse property?

For any real number x different from 0, there is a unique number, denoted by x^-1, such that x ⋅ x^-1 = 1

What is the associative law of addition?

For all real numbers x, y, and z, x + (y + z) = (x + y) + z

What is the commutative law of addition?

For all real numbers x, y, x + y = y + x

What is the associative law of multiplication?

For all real number x, y, and z, x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z

What is the commutative law of multiplication?

For all real numbers x, y, and z, x ⋅ y = y ⋅ x

What is the distributive law property?

For all real numbers x, y, and z, x ⋅ (y + z) = x ⋅ y + x ⋅ z

What is the zero factor property?

For all real numbers x and y, x ⋅ y = 0 if and only if x = 0 or y = 0 or both x and y are equal to 0

What two properties does the zero factor property consist of?

1. If x ⋅ y = 0, then x = 0 or y = 0 or both x and y are equal to 0

2. If x = 0 or y = 0, then x ⋅ y = 0

What is the set of all real numbers denoted by?

R

What is does the set of all real numbers ‘R’ include?

The integers 0, +_ 1, +_ 2, +_ 3, … and the rational numbers

What is the set of all integers denoted by?

Z

What is the set of all positive integers denoted by?

Z^+

What is the set of all negative integers denoted by?

Z^-

What is a rational number?

A ratio, or division, or two integers

What is a set of all rational numbers denoted by?

Q

What are Z^+ and Z^- a subsets of?

Z

What is Z a subset of?

Q

What is Q a subset of?

R

What is an irrational number?

A real number that is not rational; or, a real number that cannot be expressed as a simple fraction of two integers (i.e. sqrt2)

For any two real numbers x and y, one and only one of the following is true:

x < y, x = y, or x > y

For any three real numbers x, y, z, if x < y, y < z, then…

x < z

For any three real numbers x, y, z, if x < y then…

x + z < y + z

For any three real numbers x, y, z, if x < y and z > 0, then…

zx < zy

For any three real numbers x, y, z, if x < y and z < 0, then…

zx > zy

When are two equations equivalent?

If they have the same solution set; if each solution of one equation is also a solution of the second solution, and vice versa

What are the simplest equations?

Those in the forms ax + b = 0

What do we assume when we say “quadratic equation”?

The coefficient of x² is nonzero

What is the discriminant denoted by?

The Greek letter delta Δ

What does the discriminant tell us about a quadratic equation?

When it has a real solution, and if it does, whether it has one or two real solutions

When Δ = b² - 4ac > 0…

The equation has two real solutions

When Δ = b² - 4ac = 0…

The equation has only one real solution

When Δ = b² - 4ac < 0…

The equation has no real solution

What is a polynomial/

An expression in the form a_(n)x^(n) + a_(n-1)x^(n-1) +…+ a_(3)x³ + a_(2)x² +a_(1)x^(1) + a_(0), where n is a nonnegative integer, and a_0, a_1, a_2,…, a_n are real numbers with a ≠ 0

What are the coefficients of a polynomial?

a_0, a_1, a_2,…, a_n

What is a degree of a polynomial?

n, the highest power in the polynomial

What are the term of of a polynomial?

a_(n)x^(n), a_(n-1)x^(n-1),…, a_(3)x³, a_(2)x², a_(1)x^(1), a_(0)

What are monomials?

Expressions of the form x^k, where k is a nonnegative integer

When is a polynomial called the zero polynomial?

When all the coefficients of a polynomial are zeroes

When are two polynomials Q(x) and R(x) equal?

If they have the same terms

When P(x) and K(x) are two polynomials, and K(x) is not the zero polynomial, what exists?

2 unique polynomials Q(x) and R(x) such that P(x) = K(x)Q(x) + R(x); the polynomial Q(x) is the quotient, and the remainder R(x) is either the zero polynomial or of a degree that is less than the degree of K(x)

If R is the remainder when a polynomial P(x) is divided by x - a, then what does R equal?

R = P(a)

Let P(x) be a polynomial with integer coefficients. If a reduced fraction m/k is a solution to the equation P(x) = 0, then…

m divides a_0 and k divides a_n

What is the rational root theorem?

Let P(x) be a polynomial with integer coefficients. If a reduced fraction m/k is a solution to the equation P(x) = 0, then m divides a_0 and k divides a_n

What is the factor theorem?

The polynomial x - a is a factor of P(x) if and only if P(a) = 0

What is the division algorithm?

What is the expression sqrt(-1)?

‘i’

What is a complex number?

The imaginary number ‘i’, in combination with the real numbers

What is the set of complex numbers denoted by?

C

What is R a subset of?

C

What does every complex number z = a + ib correspond to?

The point (a, b)

What does i² equal?

-1

What is a conjugate?

A binomial with the same terms but an opposite sign in the middle, such as the conjugate a + bi being a conjugate of a - bi

What is the product of a complex number and its conjugate?

A real number

What is the conjugate of a complex number z denoted by?

z̄

What does the term function often indicate in everyday language?

Dependencies between quantities

What is an input?

A number that can be admitted into a function

What is an output?

An input’s corresponding number under a function

What is a domain?

The set of all possible unputs of a function

What is a range?

The set of all the outputs of a function

What is a many-to-one function?

More than one input corresponds to one output

What is a one-to-many function?

One input of a function corresponds to more than one output of the same function

What is a one-to-one function?

Two distinct inputs must correspond to two distinct outputs

When is a function invertible?

Let ‘f’ be a function. If there exists a function g such that y = f(x) if and only if x = g(y)

What is the inverse of a function denoted with (f)?

f^-1

What is the graph of f^-1 the reflection of?

f in the line y = x

What is the graph of a function ‘f’ is the set of?

All ordered pairs (x, f(x)), where x belongs to the domain of f

What is the y-intercept?

b in (0,b)

What is the x-intercept?

a in (a,0)

What is the graph of a quadratic function called?

A parabola

What is the vertex of a parabola?

The minimum or maximum point

What are the 3 properties of the case y = x²

1. The graph has a minimum at x = 0

2. The graph is symmetric

3. The function y = x² is increasing on the positive side of the x=axis and decreasing on the negative side of the x-axis

What are the 5 properties of the general case y = ax² + bx + c

1. When a > 0, the graph has a minimum at x = -b/2a

2. When a < 0, the graph has a maximum at x = -b/2a

3. The graph is symmetric with respect to the line x = -b/2a

4. When a > 0, the function is increasing on the right of x = -b/2a and decreasing on the left of x = -b/2a

5. The domain of the quadratic function is the set of R. if a > 0, the range of the quadratic function is the set of all numbers y for which y ≥ a(-b/2a)² + b(-b/2a) + c. If a < 0, the range of the function is R for y for which y ≤ a(-b/2a)² + b(-b/2a) + c

What is an exponential function?

y = a^x

What is the inverse of an exponential function called?

The logarithmic function

What is ‘a’ called in y = log_(a)x?

The base of the logarithm

What are the 5 properties of y = log_(a)x?

1. It’s equivalent to a^y = x

2. ‘a’, the base of the logarithm, must be different from 1

3. Since the function admits only positive inputs, its graph is located in the half plane to the right of the y-axis

4. There is no y-intercept

5. The function is increasing when a > 1 and decreasing when 0 < a < 1

What are natural logarithms?

Logarithms with base ‘e’

What is the inverse function of e^x?

ln(x)

If the graph of a function y = f(x), what can we construct?

y = f(x+c), y = f(x) + C, y = f(ax), y = Af(x), where a, c, A, and C are constants

What is a rational equation?

An equation that in

What is a “rational expression”?

We mean a fraction whose numerator and denominator are polynomials

What are asymptotes?

A line or curve that a function's graph approaches but never quite reaches, especially as the graph heads toward infinity

What is a power?

The expression a^n

What is ‘a’ called in a^n?

The base of the power

What is ‘n’ called in a^n?

The exponent of the power

What are the 7 properties of powers?

For any real numbers a > 0, b > 0, and a ≠ 0, and real numbers m and n…

1. a^0 = 1

2. a^m * a^n = a^(m+n)

3. a^m / a^n = a^(m-n)

4. (ab)^n = a^n * b^n

5. (a/b)^n = a^n/b^n

6. (a^n)^m = a^(n*m)

7. (when a > 0 and n is a natural number) a^(1/n) = nth sqrt(a)

What are the 8 basic properties of logarithms?

1. log_(a)1 = 0

2. log_(a)a = 1

3. a^(log_(a)b) = b

4. log_(a)b^(p) = plog_(a)^b

5. log_(a^q)b = 1/q * log_(a)b

6. log_(a)bd = log_(a)b + log_(a)d

7. log_(a)b/d = log_(a)b - log_(a)d)

8. log_(a)c = (log_(b)c)/(log_(b)a)

What is the change of base formula?

log_(a)c = (log_(b)c)/(log_(b)a)

What is the absolute value of a number?

The distance of the number from zero on the number line

What is the goal of coordinate geometry?

To describe geometric objects, such as lines and circles, algebraically through equations (and sometimes through inequalities)

What is the most basic geometric object?

The point

What is the basic idea of coordinate geometry?

The creation of a correspondence between the set of all ordered pairs of real numbers and the set of all points in the plane