Algebra & Functions Reference

Definitions

Relation

Any set of ordered pairs connecting inputs and outputs.

Example:

{(1, 2), (1, 3)}

is a relation.

Function

y = f (x)

A relation that assigns exactly one output to each permitted input.

Example:

f (x) = x^{2}

gives one

y

-value for every

x

Domain

The complete set of permitted input values.

Example: For

x

, the real domain is

x \geq 0

Range

The complete set of output values produced by a relation or function.

Example: For

y = x^{2}

, the real range is

y \geq 0

Intercept

f (x) = 0

A point where a graph crosses or touches an axis.

Example:

y = x - 3

has

x

-intercept

(3, 0)

Even function

f (- x) = f (x)

A function symmetric about the $y$ -axis.

Example:

f (x) = x^{2}

is even.

Odd function

f (- x) = - f (x)

A function symmetric about the origin.

Example:

f (x) = x^{3}

is odd.

Inverse

f^{- 1} (f (x)) = x

A relation that reverses the input-output action of a one-to-one function.

Example: The inverse of

f (x) = 2 x + 1

f^{- 1} (x) = \frac{x - 1}{2}

Composition

(f \circ g) (x) = f (g (x))

Applying one function to the output of another.

Example: If

f (x) = x^{2}

and

g (x) = x + 1

, then

f (g (x)) = (x + 1)^{2}

Asymptote

A line that a graph approaches according to its long-run or near-restriction behaviour.

Example:

y = \frac{1}{x}

has asymptotes

x = 0

and

y = 0

Hole

A removable missing point created when a restricted factor cancels algebraically.

Example:

\frac{x ^{2} - 1}{x - 1}

has a hole at

x = 1

Multiplicity

(x - r)^{m}

The number of times a factor—and therefore a zero—is repeated.

Example:

(x - 2)^{2}

has a zero of multiplicity

2

Formula sheet

9 formulas

Product of powers

a^{m} a^{n} = a^{m + n}

Variables: $a \neq = 0$ ; $m, n$ are exponents
Remember: Keep the base; add exponents.

Quotient of powers

\frac{a ^{m}}{a ^{n}} = a^{m - n}

Variables: $a \neq = 0$
Remember: Keep the base; subtract exponents.

Power of a power

(a^{m})^{n} = a^{mn}

Variables: $m, n$ are exponents
Remember: Multiply the exponents.

Negative exponent

a^{- n} = \frac{1}{a ^{n}}

Variables: $a \neq = 0$
Remember: A negative exponent indicates a reciprocal.

Log product

lo g_{b} (M N) = lo g_{b} M + lo g_{b} N

Variables: $b > 0$ , $b \neq = 1$ ; $M, N > 0$
Remember: Products become sums.

Log quotient

lo g_{b} \frac{M}{N} = lo g_{b} M - lo g_{b} N

Variables: $M, N > 0$
Remember: Quotients become differences.

Log power

lo g_{b} (M^{p}) = p lo g_{b} M

Variables: $M > 0$
Remember: Exponents move in front.

Quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Variables: $a \neq = 0$ ; $a x^{2} + b x + c = 0$
Remember: The discriminant $b^{2} - 4 a c$ determines the real-root count.

Transformation form

y = a f (k (x - d)) + c

Variables: $a, k$ : scale/reflection; $d, c$ : translation
Remember: Horizontal changes occur inside the function.

Common functions

Linear

$y = m x + b$

Domain: $R$
Range: $R$

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Quadratic

$y = x^{2}$

Domain: $R$
Range: $[0, \infty)$

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Cubic

$y = x^{3}$

Domain: $R$
Range: $R$

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Absolute value

$y = ∣ x ∣$

Domain: $R$
Range: $[0, \infty)$

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Square root

$y = x$

Domain: $[0, \infty)$
Range: $[0, \infty)$

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Reciprocal

$y = \frac{1}{x}$

Domain: $R ∖ {0}$
Range: $R ∖ {0}$

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Exponential

$y = 2^{x}$

Domain: $R$
Range: $(0, \infty)$

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Logarithmic

$y = ln x$

Domain: $(0, \infty)$
Range: $R$

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Sine

$y = sin x$

Domain: $R$
Range: $[- 1, 1]$

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Cosine

$y = cos x$

Domain: $R$
Range: $[- 1, 1]$

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Circle

$x^{2} + y^{2} = r^{2}$

Domain: $[- r, r]$
Range: $[- r, r]$

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Ellipse

$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$

Domain: $[- a, a]$
Range: $[- b, b]$

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Hyperbola

$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$

Domain: $∣ x ∣ \geq a$
Range: $R$

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Worked examples

Factor a quadratic

Factor $x^{2} - 5 x + 6$ .

Show solution

Find two numbers with product $6$ and sum $- 5$ : $- 2$ and $- 3$ .
Write $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ .
Check by expanding: $x^{2} - 3 x - 2 x + 6 = x^{2} - 5 x + 6$ .

Evaluate a composition

Let $f (x) = x^{2}$ and $g (x) = x - 3$ . Find $f (g (5))$ .

Show solution

Evaluate the inside function: $g (5) = 5 - 3 = 2$ .
Use that result as the input to $f$ : $f (2) = 2^{2}$ .
Therefore $f (g (5)) = 4$ .

Comparison table

Three quadratic forms
Form	Equation	Reveals first
Standard	$y = a x^{2} + b x + c$	$y$ -intercept; quadratic-formula coefficients
Vertex	$y = a (x - h)^{2} + k$	vertex $(h, k)$ , axis $x = h$ , opening and stretch
Factored	$y = a (x - r) (x - s)$	$x$ -intercepts $r$ and $s$ ; multiplicities

Variable glossary

$x$: independent variable or input
$f (x)$: function output at $x$
$a$: vertical scale/reflection parameter
$d, c$: horizontal and vertical translations

Student tools

Open graph studio Guided calculators

Algebra & Functions

Relation

Function

Domain

Range

Intercept

Even function

Odd function

Inverse

Composition

Asymptote

Hole

Multiplicity

y=mx+b

y=x2

y=x3

y=∣x∣

y=x​

y=x1​

y=2x

y=lnx

y=sinx

y=cosx

x2+y2=r2

a2x2​+b2y2​=1

a2x2​−b2y2​=1

Factor a quadratic

Evaluate a composition

Variable glossary

Student tools

$y = m x + b$

$y = x^{2}$

$y = x^{3}$

$y = ∣ x ∣$

$y = x$

$y = \frac{1}{x}$

$y = 2^{x}$

$y = ln x$

$y = sin x$

$y = cos x$

$x^{2} + y^{2} = r^{2}$

$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$

$\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$