Precalculus Units 1-3

What is a function in mathematics?

A relationship between two variables where one variable determines the other

In the function notation f(x) = y, what does y represent?

Dependent variable (output)

True or false: Each input in a function can have multiple outputs.

FALSE (Each input can have only one output in a function.)

To evaluate a function means to find the output for a given input by _.

plugging the input into the function expression

What is the domain of a funcation?

Set of all posabible inputs that produce a defined output

What is the range of a funcation?

The set of all outputs

What values produce undefine outputs?

Negative # such as the squereroot of a negative number.

How can we decribe domain and range?

We can use algebretic or set notation

What is the formula for average rate of change?

(y2-y1)/(x2-x1)

How do you find the 0s/x intercepts of a function

set function = 0 and solve

When is a function increasing on (a,b)

If y values are getting bigger whenever a<x<b

what are relative maxima and minima?

The y values at which the function has high points and low points.

When is a function decreasing on (a,b)

when the y values are getting smaller when A<x<b

What is a parent function?

The simplest function of a famiy of functions

Which parent functions demostrate a constart rate of change?

linear, absolute value

f(x)=x

linear function/ identity function

f(x)=lxl

absolute value function

f(x)=sqrt(x)

root or radical function

f(x)=1/x

reciprocal function

f(x)=x^3

cubic function

f(x)=x^2

quadratic function

What is the transformation and characterstics of f(x)+c?

Vertical Shift: When C>0 the graph moves UP, When C<0 the graph moves DOWN

What is the transformation and characterstics of f(x+c)?

Horizontal Shift: When C

What is the transformation and characterstics of Cf(x)?

Vertical Dilation: When C>1 the graph STRETCHES in the y-direction, When 0<C<1 (a fraction) the graph SHRINKS in the y-direction.

What is the transformation and characterstics of f(cx)?

Horizontal Dilation: When C>1 the graph SHRINKS in the x -direction, When 0<C<1 (a fraction) the graph STRETCHES in the x-direction.

What is the transformation and characterstics of -f(x)?

Reflection about x-axis: (X,Y) --> (X, -Y)

What does an equation with a Horizontal Dilation look like?

f(cx): ex, (3x²+5) (Horizontal shrink), (1/4x²+12)

Identify the function and transformation of this equation: (2x²)-3

Quadratic, Vertical Shift ( down (-3) and Horizontal dilation (2x)

What is the transformation and characterstics of f(-x)?

Reflection about x-axis: (X,Y) --> (-X,Y)

How do you test if a function f(x) is even?

An even function satisfies the property f(-x) = f(x) for all (x) in its domain.

What kind of symmetry does an even function have?

The graph of an even function is symmetric with respect to the y-axis.

How do you test if a function f(x) is odd?

An odd function satisfies the property f(-x)=-f(x) for all (x) in its domain.

What kind of symmetry does an odd function have?

The graph of an odd function is symmetric with respect to the origin (180° rotational symmetry).

How do you prove function symmetry algebraically

Substitute specific values into the equation to see if the original equation is maintained

What happens when you add or subtract functions?

You create a new function

What happens when you multiply or divide functions?

You create a new funtion

How do you determine the domain of combined functions?

Find values that satisfy the domains of the two functions that have been combined

What value of denominator would make a function undefined?

The denominator can not be 0.

True of false, function f(x) has a more restrictive domain that f(x)/g(x)

False, a function that is a combination of two functions will have a more restrictive domain.

What is a composite function?

Made up of 2 or more functions where the output of one function becomes the input of the next function

Meaning of y=f(g(x))

"f of g of x" plug x into g and that answer into f

Domain of f(g(x))

The subset of g's domain that will produce values in the domain of f

Does f(g(x))=g(f(x))

No

abbreviation for f(g(x)) and g(f(x))

(fog)(x)=f(g(x)) and (gof)(x)=g(f(x))

What is the relationship between the points on f(x) and f^{-1}(x)?

If the point (a, b) is on f(x), then the point (b, a) must be on f^{-1}(x).

inverse functions "undo " the original function

f^-1(f(c))=c f(f^-1(c))=c

What are the 3 steps to find the equation for an inverse function?

Replace f(x) with y, b) Swap x and y c) Solve for the new y and write as f^{-1}(x).

How do you find an inverse value from a table or point (e.g., find g^{-1}(12))?

Plug your inverse f^{-1}(x) into the original f(x); the result should simplify to just x.

What is the graphical relationship between f(x) and f^{-1}(x)?

They are reflections of each other across the diagonal line y = x.

If F(x) has an inverse function thats also a function. F(x) must be..

One to One

What does it mean when a function is one to one?

No two different inputs can produce the same output

How can you check if a function is One to One?

Horizontal line test

Can a function be One to One on a restricted interval of the domain?

Yes, a function can be One to One from "(A,B)" but not for all real numbers

Some points on F(x) are: (3,0) (4,2) (5,4) (6,2) Is f(x) One to One?

No, two different inputs can produce the same output

What are the steps to evaluate a piecewise function?

1. Find the interval your input belongs to (the stuff on the right) 2. Plug in the input to the equation just to the left

What is a piecewise function?

A piecewise function has different rules for certain intervals of the domian.

how do you add another rule or interval to a piecewise function?

it follows the basic structure {equation a > x ≥ b}

When does an interval have an open or closed dot?

open dot ○: when there's a > or < symbol. closed •: when there's a ≥ or ≤ symbol. e.g. 10>x≥13 makes an open dot at x=10, and a closed dot at x=13.

How do you write a quadratic in standard form?

y = ax^2+bx+c. Where c is y-intercept and "a" is vertical stretch/shrink.

How do you write a quadratic in vertex form?

y = a(x-h)^2+k. The vertex is (h,k) and "a" is vertical stretch/shrink.

How do you write a quadratic in factored/intercept form?

y = a(x-p)(x-q). Where "p" and "q" are x-intercepts and "a" is vertical stretch/shrink.

How do you find "a" and what is it?

To find "a' plug in one extra point on your parabola and solve. "a" is the vertical stretch/shrink.

How do you go from one form to another?

Go from standard form to intercept form by factoring. Go from vertex to factored form by foiling and then factoring. Go from factored form to standard by foiling.

How do you convert standard form to vertex form?

]y=a (x - k) ^2 + k where (h,k) is the vertex of the porabolar

What do you do if you get a remainder after completing the square?

If there is a remainder or some left over, there is a vertical shift

How to find the x-intercepts of your function?

Set factors to 0 and solve

How to find the y-intercept of your function?

Set x values to 0 and solve

What is multiplicity?

The number of times a factor occurs (x-p)^k

What is the graph behavior of an odd multiplicity x value?

Passes through the graph

What is the graph behavior of an even multiplicity x value?

Bounces off the graph

What does graph end behavior mean??/

the directions the ends of your graph go

Positive leading coefficient, degree even

as x -> - inf, f(x) -> inf as x -> inf, f(x) -> inf

Positive leading coefficient, degree odd

as x-> -inf, f(x)-> -inf as x-> inf, f(x)-> inf

Negative leading coefficient degree even

as x -> -inf, f(x) ->-inf as x->inf, f(x) -> -inf

Negative leading coefficient degree odd

as x -> -inf, f(x) -> inf as x-> inf, f(x) -> -inf

If (x-k) divides polynomial f(x) evenly what is one of f(x) zeros

x=k

If (x-k) divides polynomial f(x) evenly what is a factor of f(x)

(x-k)

If (x-k) divides polynomial f(x) evenly, how do you find the y-intercept

f(k)=0

If dividing polynomials gives you a remainder how do you write it

quotient+remainder/divisor

If f(5)=-2, what does that mean when dividing f(x) by (x-5)

f(x)/(x-5) has a remainder of -2

What does the fundamental Theorem of Algebra say?

A polynomial has as many roots as its highest degree.

Solve for the zeros algebraically x^2+4=0

1. x^2=-4 2. x= +- square root of -4 3. x=+-2i

What form do complex numbers take

a+bi Where a+b are real numbers and i is imaginary.

i is the square root of what? what is i^2 equal to?

1. i = square root of -1 2. i^2 = -1

If you solve using a square root, a complex solution will always have a "___ pair" what is ____ pair

A complex solution will always have a conjugate pair. A conjugate pair means that if x = a+bi is a root, x= a-bi is also a root

What are the steps to factor completely

1. Look for real zeros in a table or graph 2. Divide out known factors 3. Continue factoring if possible or use quadratic formula until all zeros are found. 4. Write remaining factors

If (2,0) is one of the zeros on a graph f(x)=x^4-2x^3+gx^2-32x+40 and has a multiplicity of 2, find remaining zeros

1. find (x-2)^2 = x^2-4x+4 2. Divide original formula by x^2 -4x+4 to find that last two zeros = x^2+2x+10 3. Use quadratic formula to find (x-1+3i) and (x-1-3i)

If a polynomial has 5 zeros what is the highest degree

5 is the highest degree

If you are finding a polynomial's zeros, and is left with 2 imaginary zeros, how to find the imaginary ones

Use Quadratic Formula

If a polynomial has a highest degree of 10 how many zeros does it have

10 zeros

What form do rational functions have?

y= f(x)/g(x) Where f and g are polynomials

How do you find a horizontal asymptote?

Compare the degree of f(x) and g(x)

What determines no horizontal asymptote?

If degree of f > degree of g

What determines a horizontal asymptote at y = 0?

If degree of f < degree of g

What determines a slant asymptote?

If degree of f is exactly 1 more than degree of g

What is the first step for graphing rational functions?

Always start by factoring

If f(a) = 0/non-zero what is a?

x = a is a zero/x-int

If f(b) = non-zero/0 what is b?

x = b is a vertical asymptote

If f(c) = 0/0 what happens on the graph?

x = c is a hole

How do you find the y value of a hole?

Plug c into simplified expression