The Building Blocks of Pre-Calc: Functions

Question 1

1) How to find out if a function is even or odd?

2) What is f (x) = –3x**2 + 4

Answer 1

1) plug –x in for x.

… if: f (–x) = f (x) then the function is even. (Result is same function as started with)

… if: f (–x) = –f (x) then the function is odd. (All signs are switched)

if. .. something else, function is neither even nor odd
2) even

f (–x) = –3(–x)2 + 4
= –3(x2) + 4
= –3x**2 + 4

Question 2

Quadratic function.

1) What is the parent function?
2) How do you move in a coordinate system?
3) The graph of any quadratic function is called?

Answer 2

1) f(x)=x**2
2) It moves 1 horizontally, up 1², over 2, up 2² …
3) Parabola

Question 3

Square-root functions.

1) What is the parent function?
2) How does it move?

Answer 3

1) g(x)=SRx

2) It moves to the right 1, up SR1; right 2, up SR2 …

Question 4

Absolute-value parent graph

1) What is the parent function?
2) What does it do to +- signs?
3) How does it move?

Answer 4

1) y = |x|
2) It turns all inputs non-­negative
3) Over 1, up 1; over 2, up 2; and on and on forever.

Question 5

Cubic function

1) What is the parent function?
2) How does it move?
3) Is the parent function even or odd? How does it infect the graph?

Answer 5

1) f(x) = x³
2) It moves right 1, up 1³; right 2, up 2³ …
3) An odd function, so rotate half of the graph 180 degrees about the origin to get the other half.

Question 6

Name the parent cube root function!

Answer 6

g(x)=3SRx # 3 isnot infront but on the SR

Question 7

1. What are reflections?
2. how to do a horizontal reflection?
3. how to do a vertical reflection? Do it with f(x) = x²

Answer 7

1. mirror image of the function either horizontal or vertical (left/right or up/down)
2. horizontal reflection= from f(x) to f(-x). A negative inside the function like in f(x)=SRx and g(x) SR-x. (Mark! NOT: -SRx).
3. vertical reflection: from f(x) to -f(x). A negative number infront of function, from f(x) = x² to g(x) = –1x²

Question 8

State and explain the expression that shows all transformations!

Answer 8

a*f[c(x-h)]+ v

a= vertical transformation

• makes the graph taller
• negative coefficeint makes function upside down
• moves over 1, up 2 · 1² (= 2) …

c= horizontal transformation
- stretchs or shrinkens a graph

h = horizontal shift

• can be added or subtracted
• negative number goes right, + left

v= vertical shift
- can be added or subtracted

Question 9

1. What are Piece-wise functions. Give an example f(x)=?

2. If you would draw that graph, you had to lift your pencil from the paper. What is this called mathematically?

Answer 9

• Functions that are broken down into several pieces. Every piece has an interval, in which only this sub-function is rueling.

Example:
f(x) = {x² – 1 if x<2}
{|x| if x> 23} # {} one big braquets pair

The first function exists only on the interval (–∞, –2]. As long as the input is less than 2, this function rules.

2. A discontinuous function

Question 10

What are rational functions?

Answer 10

Functions where the variable appears in the denominator of a fraction.

Question 11

Name the steps to find the output of rational functions!

Answer 11

1. Find vertical asymptotes
   - set the denominator =0 and solve. These point/points your graph can’t pass
2. Find horizontal asymptotes
   - count the degree.

… if denominator d > nominator d:
asymptote is x-axis (y = 0)

… num d = den d:
y= leading coefficients / leading coefficients
l.c. are the ones with the highest degrees

… num d > den d by exactly one more:

• long division of polynomials: num/den
• neglect the remainder
• quotient is the oblique asymptote.

3. Find x- and y-intercepts
   - set x = 0 for y - intercept and vice versa
4. Graphing:
   - Put in asymptotes and intercepts
   - Is graph above or under X-axis? Put in a test value of the interval for every function. If result is positive=graph above.

Question 12

How to add and subtract functions?

it is characterized by: (f + g)(x) = or (g – f)(x) =

Solve and explain what you do…

1) (x² – 6x + 1) + (3x² – 10)?
2) (3x² – 10)+(SR2x-1)
3) (3x² – 10) – (x² – 6x + 1)

Answer 12

1) I combine like terms, if existing: 4x² – 6x – 9
2) finished, because no more like terms to add.
- > so when adding functions, you only look for like terms
3) distribute the negative sign throughout the second function (sign between the functions included), and treat it like an addition:

= (3x² – 10) + (–x² + 6x – 1)
= 2x² + 6x – 11

-> when subtracting functions, use distributive property and then add functions

Question 13

How to multiply and divide functions?

it is characterized by: (fg)(x) = or (g/f)(x)=

Solve and explain what you do…

1) (fg)(x) = (x² – 6x + 1)(3x² – 10)

2)

(3x² – 10)
_______
(x² – 6x + 1)

3) How to find a specific value of a combined function. For example, (f + h)(1)?

Answer 13

1) Distribute and add:
= x²(3x²) + x²(–10) + –6x(3x²) + –6x(–10) + 1(3x²) + 1(–10).
= 3x⁴ – 10x² – 18x³ + 60x + 3x² – 10.

2) Simplify by factoring. Because neither the denominator nor numerator factor, you’re done!
3) Add functions, put 1 in for x and calculate for x.

Question 14

What does the equation (g*f)(-3) tell you to do?

Answer 14

Work from right to left (like Hebrew). Put –3 into f(x), get an answer, plug that answer into g(x)

Question 15

What is a radicant?

Answer 15

The stuff underneath the SR

Question 16

1) How to find the domain of a combined function like f(g(x))?
2) Find the domain of this combined function…

f(x)=SRx and g(x) = 25 – x², so that: f(g(x))

Answer 16

1) put g(x) function everywhere the f(x) function asks for x

2)
- g is inside f.
- f is a square root function
- Thus, the new radicand of the composed function has to be non-negative: 
- Thus: 25 – x² ≥ 0. Solve ...
- Result: x ≤ 5 and x ≥ –5,
- Conclude the domain of the combined function: 
–5 ≤ x ≤ 5.

Question 17

Inverse functions.

1) What does an inverse function do?
2) What is the inverse notation of f(x)?
3) How to graph an inverse?

4) How to find the inverse of a function? Example:
f(x)= 2x-1
_______
3

Answer 17

1) it undoes the initial function, like addition to subtraction
2) f**–1(x) #-1 is up as a power, the x not

3) Take points and switch x and y position. Why? …
… f–1(x) is a reflection of f(x). To test, put any value in f(x), like 3, and you get out -5. Put -5 in f–1(x), and you get 3.

4) Change the positions of f(x) and x. Then change f(x) to its proper name f–1(x). Then, solve equation for f-1(x)

x= 2f**-1(x)-1
_________
3

Multiply both sides with 3
Add 1
Divide by 2

f**-1(x) = 3x+1
___
2