special functions constant function ex. y = 4.95 the charlie’s smorgasbord function absolute value...
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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