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Given the toolkit function f ( x ) = x 2 , graph g ( x ) = f ( x ) and h ( x ) = f ( x ) . Take note of any surprising behavior for these functions.

Graph of x^2 and its reflections.

Notice: g ( x ) = f ( x ) looks the same as f ( x ) .

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Determining even and odd functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions f ( x ) = x 2 or f ( x ) = | x | will result in the original graph. We say that these types of graphs are symmetric about the y -axis. Functions whose graphs are symmetric about the y -axis are called even functions.

If the graphs of f ( x ) = x 3 or f ( x ) = 1 x were reflected over both axes, the result would be the original graph, as shown in [link] .

Graph of x^3 and its reflections.
(a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.

We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function .

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, f ( x ) = 2 x is neither even nor odd. Also, the only function that is both even and odd is the constant function f ( x ) = 0.

Even and odd functions

A function is called an even function    if for every input x

f ( x ) = f ( x )

The graph of an even function is symmetric about the y - axis.

A function is called an odd function    if for every input x

f ( x ) = f ( x )

The graph of an odd function is symmetric about the origin.

Given the formula for a function, determine if the function is even, odd, or neither.

  1. Determine whether the function satisfies f ( x ) = f ( x ) . If it does, it is even.
  2. Determine whether the function satisfies f ( x ) = f ( x ) . If it does, it is odd.
  3. If the function does not satisfy either rule, it is neither even nor odd.

Determining whether a function is even, odd, or neither

Is the function f ( x ) = x 3 + 2 x even, odd, or neither?

Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.

f ( x ) = ( x ) 3 + 2 ( x ) = x 3 2 x

This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.

f ( x ) = ( x 3 2 x ) = x 3 + 2 x

Because f ( x ) = f ( x ) , this is an odd function.

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Is the function f ( s ) = s 4 + 3 s 2 + 7 even, odd, or neither?


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Graphing functions using stretches and compressions

Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.

We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.

Vertical stretches and compressions

When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch ; if the constant is between 0 and 1, we get a vertical compression . [link] shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.

Questions & Answers

"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
y=x will obviously be a straight line with a zero slope
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
yes, correction on my end, I meant slope of 1 instead of slope of 0
what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
i do not understand anything
lol...it gets better
I've been struggling so much through all of this. my final is in four weeks 😭
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
thank you I have heard of him. I should check him out.
is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
What kind of answer is that😑?
I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
hello guys I'm new here? will you happy with me
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12

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