Name______Date______

Algebra II/Trigonometry Review #5

Function: relation in which each x is assigned only one y.

To test whether a relation is a function:

Ordered Pairs: x cannot repeat

Graph: vertical line test

Onto: every element in the range is being used

Examples:

1.  2. 3.

4.

Example Regents Questions:

1.  Which relation is not a function?

2.  Given which value of k will not make this relation a function?

  1. 1 b. 2 c. 3 d. 4

3.  Which graph does not represent a function?

4.  Four points on the graph of the function f(x) are shown below.

{(0,1), (1,2), (2,4), (3,8)}

Which equation represents f(x)?

(1)

(2)

(3)

(4)

Evaluating Functions:

To evaluate, plug into all values of x

Example:

Example Regents Questions

1.  If , then equals

2.  If what is the value of f(-10)?

(1) (2) (3) (4)

Composition of Functions

A way of evaluating a function where the answer from evaluating one function is plugged into the second function.

**always work from right to left(from inside to outside)

Notation:

Example #1: Example #2:

Example Regents Equations

1.  If and , what is the value of

(1) -13 (2) 3.5 (3) 3 (4) 6

2.  If and then g(f(x)) is equal to

(1) (2) (3) (4)

3.  If and , determine the value of

4.  If and , find f(g(x)).

Domain and Range

Domain: the x values of the function. Find the points farthest to the left and right and describe the set using the x values

Range: the y values of the function. Find the highest and lowest points and describe the set using y values.

Example #1: #2: #3:

Example Regents Questions:

1.  What is the domain of the function shown below?

2.  What is the domain of the function shown below?

Domain Part II

Domain Issue / How to find Domain: / How to Represent Domain:
Rational / Set Denominator equal to Zero /
Radical / Can’t have negative under radical
Set radicand greater than or equal to 0 /

Radical in Denominator / Can’t have negative or zero
Set radicand greater than zero /

Example #1: Example #2: Example #3:

Example Regents Questions:

1.  For , what are the domain and range?

  1. and
  2. and
  3. and
  4. and

2.  The domain of is the set of all real numbers

  1. Greater than 2
  2. Less than 2
  3. Except 2
  4. Between -2 and 2

3.  If , what is the domain of the function?

  1. Domain: ; range:
  2. Domain: ; range:
  3. Domain: ; range:
  4. Domain: ; range:

Inverse Functions:

The inverse of a function is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. (switch x and y)

One-to-one: each second element corresponds to one and only one first element.

--ordered pairs: x’s don’t repeat, y’s don’t repeat.

--horizontal line test: draw a horizontal line if it hits the graph only once its one-to-one

--The original function must be one-to-one in order for its inverse to be a function

Examples: Determine if the following functions are one-to-one:

1.  2.

Swap ordered pairs: switch x and y

Given the function, f, find the inverse. Is the inverse also a function?

Algebraically: 1. Set the function = y

2.  Switch x and y

3.  Solve for y

Find the inverse of each of the following functions

1.  2.

Example Regents Questions

1.  Which graph represents a one-to-one function?

2.  Which function is one-to-one?

  1. b. c. d.

3.  Which diagram represents a relation that is both one-to-one and onto?

4.  If find