Functions And TRANSLATIONS

Question 1

What is the sqaure root function

Answer 1

This is f x = sqaure root but only POSITVE square root, remember function is something to one, thus can’t be -

Here this is also a ONE TO ONE FUNCTION

Question 2

F (x) = +-root (x2 +1),
Is this a function?
What kind of mapping

Answer 2

Not a function , as many outcomes
Many to many, as -1 + 1 gives same

Question 3

Remember in transformations for x axis, anything you do goes IN THE BRACKETS

Answer 3

therefore if tryna got sin (2x-1) I could +1 ( x-1) and then stretch 1/2 = sin (2(x) - 1). Each transformation replaces X with the trandfomstjkn, NKT THE WHOLE THING

if I did the stretch first, it would be sin (2x) . Now if I want -1, as it goes INSIDE THE BRACKET, I must translate by 1/2 parallel to x axis

= sin 2(x-0.5) = 2x -1 as required

So don’t lack, always goes inside the bracket, can achieve thing TWO WAYS!

Question 4

Rememebr if I apply a transformation to x or y what to do about values?

Answer 4

Reverse, want to stretch 2= 1/2, add 1 = -1, but for y it’s as expected

Therefore finding time period of sin (kx +1), as the 1 shifts the position but not the actual period, this is a stretch 1/k. And as period normally 2Pi , 2Pi/k!

Question 5

What about y, if I want 2 f(x) -1 how to do?

Answer 5

Either stretch by 2 then -1
Or -1/2 and THEN STRETCH BY 2
As this gives 2 f(x) -1 as required

So the y is the ENTIRE EQUATION AS A BRACKET
so yes it does matter
-but know you can get to the same thing in two different paths

Question 6

Easiest way to apply the transformations?
Changed for x and y, explain

Answer 6

For y if you do the outside first, then stretch will stretch thid too, hence stretch then outside

For x if you do stretch , then outside, the outside will get stretched, so thr stretch and then outside

Very easy, just make sure to use REPLACING with brackets for x bs brsckets of the entire equation for y

Question 7

And in general where does the y stay the same come from?

Answer 7

It acc doesn’t, you apply the reverse transformation to the y part of the thing. And then when you rearrange it works out to be the “same”
But if stuck yh try this out

Remember x brackets and y transformation in brackets too, do stretch first and you’re good

Question 8

And if the translations ar edifrernt x or y, does order matte

Answer 8

No order doesn’t matter if apply to two separate !

Question 9

F 2 (x)?

Answer 9

This is f of of f x don’t lack

Question 10

Inverse function rules

Answer 10

Is a reflection in y =x
Where y values swapped to x

Now must be 1 to 1, so restrict

And the domain and ranges are swapped too

Question 11

Remember how to convert from mod inequality

Answer 11

Add , divide by 2 and subtrsvt from x

Now subtrsvt from right snd left, should get to same number, and so take the bigger one!

Question 12

How does the negative part of the modulus lok

Answer 12

Remember mod makes EVERYTHING POSITIVE , so negative modulus would be negative x all of it

So when drawing grsdient it’s the negative part

Question 13

How to solve the mod equations using algebra, what to do

Answer 13

Must sqaure and find roots, and then CHECK IF THESE WORK
The only ones that will work are the ones that work

Remember to put it in the y equation after, as this will guarantee correct ( else have to draw negative mode etc to see)

Using graoh, draw them out and use negative versions and intersect

Question 14

How to do the inequalities? Can you use algebraic? What to remember again

Answer 14

Can use algebraic but ince again MUST check which roots satisfy equation. If both do then use normal inequality under over the graoh approach

If only one satisfies then ignore the other, and have this one

Use graph method anyways to check!