Function V

Front 1

Function (y as func of x) or not?

x³+y²= 1

Back 1

No

y= ±¬(1-x³)

Front 2

Function (y as func of x) or not?

x²+y³=1

Back 2

Yes

y=³¬(1-x²)

Front 3

Function (y as func of x) or not?

x²y = 1-3y

Back 3

no b/c x^(½)

Front 4

Function (y as func of x) or not?

{(-2,y)|-3<4}>

Back 4

No, since x=-2

Front 5

Function (y as func of x) or not?

> if Y indicate D&R

{(x,3)|-2<4}>

Back 5

Yes

D: (-2,4)

R: {3}

Front 6

Function (y as func of x) or not?

> if Y indicate D&R

{(x,y)|x is an odd integer and y is an even integer

Back 6

No (not necessarily), describes all points in Quad II

Front 7

Function (y as func of x) or not?

> if Y indicate D&R

{(x,1)|x is an irrational number}

Back 7

Yes

D: {x|x is an irrational number}

R: {1}

Front 8

Function (y as func of x) or not?

> if Y indicate D&R

{(x,x²)|x is a real number}

Back 8

Yes

since y=x²

D: (-~, ~)

R: [0,~)

Front 9

Function (y as func of x) or not?

> if Y indicate D&R

{(x²,x)|x is a real number}

Back 9

No, b/c y=±¬x

Front 10

Function (y as func of x) or not?

> if Y indicate D&R

x^2= 9-y^2

Back 10

y=±√(9-x^2)

.*. no

x^2+y^2= 9 produces graph of square

Front 11

Find domain of func

f(x)= √(4-x)/(x-1)

Back 11

D: (-∞,1)U(1,4]

Front 12

Let f(x) = 2-3x+x^2

find f(x^2)

Back 12

f(x^2)= 2-3x^2+x^4

Front 13

Let g(x)= (x+4)/(2x-1)

find g(x+1)

Back 13

g(x+1)= (x+5)/(2x+1)

Front 14

How find x-ints & y-ints from equation?

Back 14

(i) x-ints: set y=0, then solve x

ex. f(x)=2-3x

0= 2-3x

x= 2/3, .*. (2/3, 0)

(ii) y-ints: set x=o, then solve y

ex. f(x)= 2-3x

y= 2-0= 2

.*. (0,2)

Front 15

Find intercepts of x^2-5x+y^2+3y

Back 15

(i) x-int:

x^2-5x=0

x(x-5)=0

.*. (0,0) & (5,0)

(ii) y-int:

y^2+3y=0

y(y+3)= 0

.*. (0,0) & (0,3)

Front 16

Codomain vs range

Back 16

(i) Codomain= What may possibly come out func

(ii) Range= What actually comes out func

Front 17

Range of g(x) = x³-8

Back 17

R (x³ can take any real #)

Front 18

Find D & R of func

Back 18

D: [-3,3)

R: [-4,5]

Front 19

Find D & R

Back 19

D: [-2,-1] U [0,4)

R: [1,5)

Front 20

Back 20

Front 21

f(x) = (x²-9)/(x²-4)

Find domain of f and any x or y ints

Back 21

(i) D: x²=¬4=±2;

(-~,-2)U(-2,2)U(2,~)

(ii) xint: 0 = (x²-9)/(x²-4)

0= (x-3)(x+3)/(x+2)(x-2);

(3,0)&(-3,0)

yint: y=(0-9)/(0-4)

y= 9/4; (0,9/4)

Front 22

Difference Quotient

Back 22

[F(x+h)-f(x)]/h, h≠0

Front 23

Find diff quotient of f(x)=2x²-4x+3

Back 23

[F(x+h)-f(x)]/h, h≠0

(i) f(x+h) = 2x²+4xh+2h²-4x-4h+3

(ii)[F(x+h)-f(x)]= h(4x+2h-4)

(iii) [F(x+h)-f(x)]/h,= 4x+2h-4

Front 24

F

Back 24

F

Front 25

D

Back 25

D

Front 26

How do you vertically shift a function? (ii)

Back 26

f(x) + k or f(x) - k

Add (shift each point up) or Subtract (shift each point down) "k" (assuming pos) from f(x)

.*. increasing or decreasing each value of y by k value

Front 27

How do you horizontally shift a function? (ii)

Back 27

f(x+h) or f(x-h)

Add (shift each point left) or subtract (shift each point right) "h" (assuming pos) to x in f(x).

Front 28

(i) Graph f(x) = √x; Plot at least three points. (ii) graph g(x) = √x)− 1

(iii) graph j(x) = √x−1) (iv) graph m(x) = √x+3) − 2

Back 28

(ii) g(x) = √x)− 1: subtract 1 to f(x),.*. shift each y value down 1 unit

[y= f(x)-1] (iii) graph j(x) = √x−1): subtract 1 to x; shift each x value right 1 unit

[y= f(x-1)] (iv) graph m(x) = √x+3) − 2: subtract 2 to f(x) & add 3 to x;

shift each y value down 2 units & each x value left 3 units

[y=f(x+3)-2]

Front 29

How to reflect function across y & x-axis?

Back 29

(i) X-axis: y= -f(x): multiply each y coord w/ -1

(ii)Y-axis: y= f(-x): multiply each x coord w/ -1

Front 30

Graph g(x) = √-x) & give domain + range

Back 30

(i) graph g(x) = √x)

(ii) reflect g(x) = √x) across x-axis

(iii) D: (−∞, 0]

R: [0,∞)

Front 31

Graph j(x) = √3−x), give D&R

Back 31

(i)Horiz Shift: graph j(x) = √x & shift each x-coord 3 units left [y= f(x+3)]

(ii) Reflect about y-axis: graph j(x) = √-x+3) by making each x-coord negative

(iii) D: -x+3≥0, x≤3; (-∞,3]

R: [0,∞)

Front 32

Graph: m(x) = 3 − √x) & give D+R

Back 32

(i) Vertical Shift: graph m(x)= √x) & add 3 to each y-coord [y=f(x)+3]

(ii) Reflect about x-axis: graph m(x) = 3 − √x) by multiplying each y coord by -1.

(iii) D: [0,∞), R: (-∞,3]

Front 33

How do you vertically scale (stretch & shrink) a function?

Back 33

(i) Vertical Stretch: graph y= af(x) if a>0: multiply each y coord by "a"

(ii) Vertical Shrink: graph y= af(x) if 0<a<1 : multiply each y coord by "a"

Front 34

How do you horizontally scale (stretch & shrink) a function?

Back 34

(i)Horizontal Stretch: graph y= f(bx) if 0<b<1: multiply each x coord by b

(ii) Horizontal Shrink: graph y= f(bx) if b>0 : multiply each x coord by "1/b"

Front 35

Graph g(x) = 3√x), give D&R

Back 35

(i) draw g(x) = √x)

(ii) Vertical Stretch: multiply each y coord by 3 [y=af(x)]

(iii) D&R: [0,∞)

Front 36

Graph j(x) = √9x), give D&R

Back 36

(i) graph j(x)=√x

(ii) Horizontal Compress: graph j(x) = √9x by multiplying each x-coord by 1/9

(iii) D&R: (0,∞)

Front 37

Graph m(x) = 1−√[x+3]/2)

Back 37

C

Front 38

Transformations

Suppose f is a func

g(x) = Af(Bx + H) + K, A≠0 B≠0

How do you graph this func?

Back 38

(i) Add H to each x coord: if H>0 shift each left & if H<0 shift right

(ii) Divide each x coord by B: if 0<B<1 Horiz stretch or if B>1 Horiz compress; if B negative reflect y-axis

(iii)Add K to each y-coord: if K>0 shift each up & if K<0 shift down

(iv) Divide each y coord by A: if 0<A<1 vert compress or if K>1 vert setretch; if A negative reflect x-axis

Front 39

When's a func increasing?

Back 39

when y-value increases as x-value increases

when x1 < x2 then f(x1) ≤ f(x2)

Front 40

Strictly Increasing Func

Back 40

Increasing func w/ no flat sections

when x1 < x2 then f(x1) < f(x2)

Front 41

Usually we are only interested in some interval, like this one:

Back 41

This function is increasing for the interval shown(it may be increasing or decreasing elsewhere)

Front 42

When's a func decreasing?

Back 42

When y values decrease as x values increase

when x1 < x2 then f(x1) ≥ f(x2)

>strictly decreasing: when x1 < x2 then f(x1) > f(x2)

Front 43

Constant Func (graphically)

Back 43

Horizontal line (zero rise)

Front 44

Y= mx+b

When is a line...

(i) increasing

(ii) decreasing

(iii) constant

Back 44

(i) m>0

(ii)m<0

(iii) m=0

Front 45

"injective" or "one-to-one" func

Back 45

Never get same "y" value twice.

>useful b/c can be reversed, can go from "y" back to "x" value (can't do when >1 possible "x" value)

Front 46

Even Func

Back 46

When f(x) = f(-x) for all x

> substitute f(-x) for for f(x) & see if get same result

ex. f(x) =x²; f(2)=4 & f(-2)=4

called "even" fun b/c funcs x², x⁴, x^6 x^8 etc behave like this.

>But even exp doesn't always make even function, ex. (x+1)²

Front 47

Odd Func

Back 47

−f(x) = f(−x) for all x

> orgin symmetry

> substitute -f(x) for for f(-x) & see if get same resultex. f(x) =x³; f(-3)= -27 & -f(3)= -27

called "odd" b/c funcs x³, x^5 etc behave thus>But odd exp doesn't always make odd function, ex. x³+1

Front 48

Don't be misled by the names "odd" and "even" ... they are just names ... and a function does not have to be even or odd.

Back 48

In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this:

Front 49

Only func that's even & odd?

Back 49

If f(x)= 0

Front 50

Special Properties (Adding Funcs)

(i) Sum of 2 even funcs is even/odd

(ii) Sum of 2 odd funcs is even/odd

(iii) Sum of even & odd func is?

Back 50

(i) even func

ex. F(x) + G(x) = x²+x⁴

(ii) odd func

(iii) neither even nor odd (unless one function is zero).

Front 51

Special Properties (Multiplying Funcs) (i) Product 2 even funcs is even/odd function.(ii) Product 2 odd funcs is even/odd function.(iii) Product even func & odd func even/odd function.

Back 51

(i) even

(ii) even

ex. x^3*x^5= x^8

(iii) odd

ex. x²*x³= x^5