Special Functions

• Constant Function ex. y = 4.95 The Charlie’s Smorgasbord Function

• Absolute Value Function

ex. y = | x | absolute Value

• Step (Greatest Integer) Function ex. y = [ x ] The Bob Barker “Price is Right” Function

• Piece-wise Function ex. Made up of pieces!

Example 1 Step Function

Example 2 Constant Function

Example 3 Absolute Value Functions

Example 4 Piecewise Function

Example 5 Identify Functions

Psychology One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this solution.

Explore The total charge must be a multiple of $85, so the graph will be the graph of a step function.

Plan If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is $85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is $170, and so on.

Solve Use the pattern of times and costs to make a table, where x is the number of hours of the session and C(x) is the total cost. Then draw the graph.

425

340

255

170

85

C(x)x

Answer:

Examine Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint.

Sales The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation.

Answer:

Graph

For every value of The graph is a horizontal line.

g(x) = –3

x g(x)

–2 –3

0 –3

1 –3

0.5 –3

Answer:

Graph

Answer:

Graph and on the same coordinate plane. Determine the similarities and differences in the two graphs.

Find several ordered pairs for each function.

x | x – 3 |

0 3

1 2

2 1

3 0

4 1

5 2

x | x + 2 |

–4 2

–3 1

–2 0

–1 1

0 2

1 3

Graph the points and connect them.

Answer:• The domain of both graphs

is all real numbers.

• The range of both graphs is

• The graphs have the same shape, but different x-intercepts.

• The graph of g (x) is the graph of f (x) translated left 5 units.

Graph and on the same coordinate plane. Determine the similarities and differences in the two graphs.

• The domain of both graphs is all real numbers.

• The graphs have the same shape, but different y-intercepts.

• The graph of g (x) is the graph of f (x) translated up 5 units.

Answer:

• The range of is The range of

is

Step 1 Graph the linear function

forSince 3 satisfies this inequality, begin with a closed circle at (3, 2).

Graph Identify the domain and range.

Step 2 Graph the constant function

Since x doesnot satisfy this inequality, begin with an open circle at (3, –1) and draw a horizontal ray to the right.

Graph Identify the domain and range.

Graph Identify the domain and range.

Answer: The function is definedfor all values of x, so the domainis all real numbers. The valuesthat are y-coordinates of points on the graph are all real numbersless than or equal to –2, so therange is

Graph Identify the domain and range.

Answer: The domain is all realnumbers. The rangeis

Determine whether the graphrepresents a step function, a constant function, anabsolute value function, or a piecewise function.

Answer: Since this graphconsists of differentrays and segments, it is a piecewisefunction.

Determine whether the graphrepresents a step function, a constant function, anabsolute value function, or a piecewise function.

Answer: Since this graph is V-shaped, it is an absolute value function.

Determine whether each graph represents a step function, a constant function, an absolute value function, or a piecewise function.

a. b.

Answer: constant function

Answer: absolute valuefunction