#2 Functions

Function

A mapping/matching from a set of inputs to a set of outputs according to some kind of rule where every input has exactly one output. 

There are one-to-one functions and many-to-one functions.

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Representations:

Graphical

Algebraic

Numerical

Verbal

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In a function for y as a function of x the variable x IS THE INDEPENDENT VARIABLE (Because you can change it) and y is THE DEPENDANT VARIABLE (Because it depends on x’s value)

Vertical Line test

The test consists on drawing a vertical line through the graph pf a function. If at any point the line crosses through more than one point in the function it ISN’T a function as it is one-to-many (NOT A FUNCTION)

Horizontal Line test

Test consists of drawing a horizontal line through the graph of a function. If the line crosses through more than one point of the function then it’s not one-to-one as that makes the function many-to-one.

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\*\*\*Many-to-one functions can’t have inverse functions unless their domain is restricted as they would become one-to-many which aren’t functions.

Domain

The set of all possible input values for the function

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Range

The set of all output values created for the function

Domain & Range Notation

Even Functions

Function is that if, for all x in the domain, -x is the domain and f(x) = f(-x). A functions even if the graph is SYMETRICAL ABOUT THE Y-AXIS.

Odd Functions

Function is that if, for all x in the domain, -x is in the domain and f(-x) = -f(x). The graph of an odd function has ROTATIONAL SYMMETRY OF 180⁰ about the origin.

Operations With Functions & Domain (Addition, Subtraction, Product)

The domain of a function that consists of more than one function would be the SET THAT IS COMMON TO ALL FUCNTIONS.

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Ie. D₁∩D₂

Operations With Functions & Domain (Quotients)

The set that is common to all the functions and any values that make the DENOMINATOR ZERO would be EXCLUDED FROM THE INTERSECTION OF THE SET.

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D₁∩D₂ − {x|g(x) = 0}

Sum/Difference &/of Even & Odd Functions

Even ± Even = Even

Odd ± Odd = Odd

Even ± Odd = Neither

Product of Even &/or Odd Functions

Even \* Even = Even

Odd \* Odd = Even

Even \* Odd = Odd

Domain Of Composite Functions

the domain of the composite function f(g(x)) is made up of all the x values in the DOMAIN OF g whose images g(x) belong to the domain of f.

The domain of a composite is all the x values that can go in the INSIDE FUNCTION which comply with the domain of the OUTSIDE FUNCTION.

Basically, all the x values from the inside function that give outputs within the domain of the outside function.

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When dealing with functions different functions such as ones that include radicals a similar process can be followed but keep in mind that the exceptions/exclusions from the domain may be inequalities (ie. a range of values such as y≥4 or wtv).

Range of Composite Functions

The values for the domain of the composite are then mapped as a function of f (the outside function) into it’s range, producing the range for f(g(x)), which is a SUBSET of the RANGE of f.

When the new domain is ‘open’ the range of the outside function doesn’t change. ‘Open’ I mean that as the new domain goes to infinity from the left and right outputs keep appearing.

(ie Domain has open brackets)

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When the new domain is ‘Closed’ the range of the outside function becomes restricted to a new range. By ‘Close’ I mean that the new domain is between some values and doesn’t go to infinity from either left or right.

(ie. New domain only allows for inputs between {x∈| \[3,10\] }

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When dealing with different functions such as ones that include radicals a similar process can be followed but keep in mind that the new ranges may only be valid between a specific range of values.

Decomposing Composites

Given a composite function f(g(x)) where f(x) is known, then g(x) can be obtained by setting f(g(x)) = f(x) and solving for g(x)

Inverse Functions

Two functions are inverses to each other if their input-output pairs are reversed, so that if one function takes a value as an input and gives a second value as an output, then the other function takes the second value as the input and gives the first value as an output.

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Formally → If functions f and g satisfy the property:

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f(x) = y ↔ g(y) = x

Then you call function g(x) the inverse of f(x)

g(x) = f⁻¹(x)

Inverse Function Graphical Representation

The graph of the inverse function can always be created by reflecting the graph of the original function across the line y = x.

Otherwise stated, a function and its inverse are always symmetrical across the line y = x.

Inverse Function Domain & Range

The domain of an inverse function is the same as the range of the original function. The range of an inverse function is the same as the domain of the original function.

Existence of an Inverse Function

A function f has an inverse function is an only if the function is a one-to-one function. The graph of f PASSES both the VERTICAL LINE TEST and the HORIZONTAL LINE TEST for all values of x within the domain.

If the function does NOT PASS the HORIZONTAL LINE TEST its domain can be restricted to allow the inverse function to exist within the restricted domain.

Proving that two Functions are Inverses

Two functions, f(x) and g(x) are inverses if:

f\[g(x)\] = x = g\[f(x)\]

Self Inverses

A self inverse is a function where f(x) = f⁻¹(x)

Since f(x) = f⁻¹(x)

→ f(f⁻¹(x)) = f⁻¹(f(x)) = x

also

→ f(f(|x|)) = f⁻¹(f⁻¹(x)) = x

Identity Function

A line of all points where the x and y coordinates are identical. The Identity function maps each input to itself. The identity line is the function y = x.

Parent Function (Constant)

Parent Function (Quadratic)

Parent Function (Absolute value)

Parent Function (Cube Root)

Parent Function (Inverse Rational)

Parent Function (Exponential)

Parent Function (Linear)

Parent Function (Cubic)

Parent Function (Square Root)

Parent Function (Reciprocal)

Parent Function (Logarithmic)

Parent Function (Greatest Integer)

Parent Function (Sine)

Parent Function (Cosine)

Parent Function (Tangent)

Transformations g(x) = kf(x)

Vertical stretch by a factor of k

Transformations g(x) = f(kx)

Horizontal stretch by a factor of 1/k

Transformations g(x) = -f(x)

Reflection across the x-axis

Transformations g(x) = f(-x)

Reflection across the y-axis

Vector Notation

Order of Applying Transformation

First a function must be put into standard form (Where “x” isn’t multiplied by any factor). Once the function is reorganized you first apply the horizontal/vertical stretches and then the translations.

Transformations g(x) = (f(x))²

The function becomes bounty and all the negative y-values get reflected over the x-axis.

* The points with y-coordinates 0 or 1 are invariant (don’t change)
* The graph y = \[f(x)\]² only touches the x-axis at it’s x-intercepts
* the graph y = \[f(x)\]² lies above or on the x-axis for all x
* The vertical Asymptotes for y = f(x) are also the vertical asymptotes for y = \[f(x)\]²
* When -1

Transformations g(x) = |f(x)|

All the negative y-values are reflected over the x-axis

Transformations g(x) = f(|x|)

The graph to the right side of the y-axis is mirrored in the left side of the y-axis

Transformations g(x) = 1/f(x)

* If the order pairs (a.b) belong to y = f(x) then (a, 1/b) is the reciprocal
* Reciprocal of 0 is undefined therefore, where f(x) = 0 there are vertical asymptotes for the reciprocal
* The reciprocal of ±1 is ±1, therefore the reciprocal function shares the points on the original function where y = ±1
* If b is the y-int then 1/b is the new y-int for the reciprocal
* The minimum values of f(x) will occur at the same x-values as the maximum values of 1/f(x) and vice versa
* As f(x)→0, 1/f(x)→∞, conversely, as f(x)→±∞, 1/f(x)→0

Absolute Value

The absolute value or modulus of a real number x is it’s distance from 0 on the number line.

Formal Definition: (Picture)

Properties of Absolute Value #1

|x|≥0

Properties of Absolute Value #2

|x|² = x²

Properties of Absolute Value #3

|x/y| = |x| / |y|

Properties of Absolute Value #4

|-x| = |x|

Properties of Absolute Value #5

|x *y| = |x|* |y|

Properties of Absolute Value #6

|x - y| = |y - x|

Evaluating Absolute Value Inequalities

When solving for inequalities by squaring both side, evaluate the zeros before assigning them an inequality sign.

When Solving inequalities case by case and you get two contradicting answers evaluate them secretly to see which is correct.

2 x 2 System Conclusions (Unique Solution)

The lines intersect at a point. there is a unique ordered pair, (x,y), for the variables.

2 x 2 System Conclusions (Infinite number of solutions)

The lines intersect in a line producing an infinite set of intersection points. There are infinitely many ordered pair, (x,y), for the variables. Algebraically the x and y variables disappear and we are left with a true statement.

2 x 2 System Conclusions (No Solution)

The lines do not have any intersection points. There are no values for the variables that satisfy both equations in the system. Algebraically the x and y variables disappear and we are left with a false statement. Graphically the two lines are parallel but in different places, therefore they never intersect.

Consistent System

A system is called consistent if it has a unique solution or infinitely many solutions.

Inconsistent System

A system is called inconsistent if it has no solution.

Augmented Matrices

Row Echelon Form

Three Things When Solving Systems

1) Get the zero in the bottom corner first and then get the zero in the first spot of the middle row (Preferably in the same step)

2) Then get the last zero by using the third and second row

3) Solve for each variable

TIPS for Solving Systems #1

When showing word/operations write the row you are working on to the right of the operation.

TIPS for Solving Systems #2

When doing multiple operations you can’t have operations repeating left sides.

TIPS for Solving Systems #3

Always use the second row to work on the third row for obtaining the final zero.

Vertex/Completed Square Form

f(x) = a(x-h)² + k →→ Vertex (h,k)

Standard Form

f(x) = ax² + bx + c →→ y-int (0,c)

Intercept/Factored Form

f(x) = a(x-p)(x-q) →→ x-int, (p,0) , (q,0)

Completing The Square

Factoring Magic Box

Difference of Squares

(a+b)(a-b) = a² - b²

Perfect Square Trinomial

(a±b)² = a² ± 2ab + b²

Grouping

Sum of the Roots

S.O.R = -b/a

Product of the Roots

P.O.R = c/a

Polynomial Functions

Functions with multiple terms given that all the powers of the variables must be positive integers and all of the coefficients must be real.

Characteristics of Polynomial Functions (3)

* The graph of a polynomial function of degree “n” has at most (n-1) turning points
* The highest (maximum) number of zeros a polynomial can have corresponds with the highest degree
* Something about polynomials and their degrees is that the highest degree of a polynomial corresponds with the number of times you need to get the difference between the integer values of x to have a constant difference

Long Division

Reverse Tabular Division

Synthetic Division/ Horner’s Algorithm

This method ONLY works for LINEAR FACTORS

Remainder Theorem

Any polynomial p given by p(x) = axⁿ +bxⁿ⁻¹ + … + cx + d can be written in the form:

(x-h)(something) + Error

Where the error term equals the value of the polynomial at x = h

Factor Theorem

In particular, if p(h) = 0, that is, x = h is a zero of the polynomial; then the error term is zero and we have that (x-h) is a factor of the polynomial.

p(x) = (x-h)(something)

How to Rewrite a Polynomial as a product of a linear factor and a constant (Using the remainder theorem)

Given a Polynomial p is being divided by something which can be written as (x±h) you can keep factoring (x±h) from the polynomial until there is only a coefficient remains. This coefficient is the remainder of the polynomial when divided by (x±h) . The remainder is equivalent to plugging in “h” into the polynomial as “x”.

Rational Root Theorem

Assuming that the roots of a polynomial is a rational number p/q, p,q∈Z and share no common factors then the roots can be obtained with this method.

Fundamental Theorem of Algebra

A polynomial of degree n can be written as a product of n factors.

Conjugate Root Theorem

Factor Sum of Cubes

Factor Difference of Cubes

Sum and Product of the Roots of Polynomial Equations

Graphing Polynomial

* Leading coefficient test
* Find the zeros
* Assess Multiplicity
* Sketch the function

Leading Coefficient Test/ End Behavior (Odd with positive)

Odd function with positive coefficient starts down and ends up

Leading Coefficient Test/ End Behavior (Odd with negative)

Odd function with negative coefficient starts up and ends down

Leading Coefficient Test/ End Behavior (Even with positive)

Even function with positive coefficient starts up ends up

Leading Coefficient Test/ End Behavior (Even with negative)

Even function with negative coefficient starts down ends down

Finding the Equation of the Polynomial

* Identify the number of different roots “n”
* Identify whether there are any equal roots. if so look at the shape at the equal root to identify the factor for this root
* Do not forget to find the leading coefficient using a given point on the graph
* Identify end behavior

Types of Rational Functions for f(X) = g(x)/h(x) (g>h)

When the power of the numerator is greater than the denominator by more than one power there is no horizontal asymptote

Types of Rational Functions for f(X) = g(x)/h(x) (g

When the power of the numerator is less than the power of the denominator, the horizontal asymptote is equal to 0

Types of Rational Functions for f(X) = g(x)/h(x) (g=h)

When the power of the numerator and denominator are the same the horizontal asymptote is the ratio of the leading coefficient for each function

Possible Rational Functions Shapes