Special Functions Constant Function ex. y = 4.95 The Charlie’s Smorgasbord Function Absolute Value Function ex. y = | x | absolute V alue Step (Greatest Integer) Function ex. y = [ x ] The Bob Barker “Price is Right” Function Piece-wise Function ex. Made up of pieces!

Lesson 6 Contents Example 1Step Function Example 2Constant Function Example 3Absolute Value Functions Example 4Piecewise Function Example 5Identify Functions

Example 6-1a 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.

Example 6-1b 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 C(x)x

Example 6-1c 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.

Example 6-1d Sales The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation. Answer:

Example 6-2a Graph For every value of The graph is a horizontal line. g(x) = –3 xg(x)g(x) –2– –3 Answer:

Example 6-2b Graph Answer:

Example 6-3a Graphandon the same coordinate plane. Determine the similarities and differences in the two graphs. Find several ordered pairs for each function. x| x – 3 | x| x + 2 | –42 –31 –20 –

Example 6-3b 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.

Example 6-3c Graphandon 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 isThe range of is

Example 6-4a Step 1Graph the linear function for Since 3 satisfies this inequality, begin with a closed circle at (3, 2). GraphIdentify the domain and range.

Example 6-4b Step 2Graph the constant function Since x does not satisfy this inequality, begin with an open circle at (3, –1) and draw a horizontal ray to the right. GraphIdentify the domain and range.

Example 6-4c GraphIdentify the domain and range. Answer: The function is defined for all values of x, so the domain is all real numbers. The values that are y -coordinates of points on the graph are all real numbers less than or equal to –2, so the range is

Example 6-4d GraphIdentify the domain and range. Answer:The domain is all real numbers. The range is

Example 6-5a Determine whether the graph represents a step function, a constant function, an absolute value function, or a piecewise function. Answer:Since this graph consists of different rays and segments, it is a piecewise function.

Example 6-5b Determine whether the graph represents a step function, a constant function, an absolute value function, or a piecewise function. Answer:Since this graph is V-shaped, it is an absolute value function.

Example 6-5c 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 value function