1. piecewise-defined function
piecewise-linear function
step function
greatest integer function
absolute value function
Vocabulary

2. Piecewise-Defined Function
Step 1 Graph the linear function f(x) = x – 1 for x ≤ 3. Since 3 satisfies this inequality, begin with a closed circle at (3, 2).
Example 1

3. Piecewise-Defined Function
Step 2 Graph the constant function f(x) = –1 for x > 3. Since x does not satisfy this inequality, begin with an open circle at (3, –1) and draw a horizontal ray to the right.
Example 1

4. Piecewise-Defined Function
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 {f(x) | f(x) ≤ 2}.
Example 1

5. A. domain: all real numbers range: all real numbers
B. domain: all real numbers range: {y|y > –1}
C. domain: all real numbers range: {y|y > –1 or y = –3}
D. domain: {x|x > –1 or x = –3} range: all real numbers
Example 1

6. Write the piecewise-defined function shown in the graph.
Write a Piecewise-Defined Function
Write the piecewise-defined function shown in the graph.
Examine and write a function for each portion of the graph.
The left portion of the graph is a graph of f(x) = x – 4. There is a circle at (2, –2), so the linear function is defined for {x | x < 2}.
The right portion of the graph is the constant function f(x) = 1. There is a dot at (2, 1), so the constant function is defined for {x | x ≥ 2}.
Example 2

7. Write the piecewise-defined function. Answer:
Write a Piecewise-Defined Function
Write the piecewise-defined function.
Answer:
Example 2

8. Identify the piecewise-defined function shown in the graph.B. C. D.
Example 2

9. Use a Step Function
PSYCHOLOGY One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation.
Understand 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 3

10. Use a Step Function
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.
Example 3

11. Use a Step Function
Answer:
Check 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 3

12. SALES The Daily Grind charges $1
SALES The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation.
A. B.
C. D.
Example 3

13. Concept

14. Graph y = |x| + 1. Identify the domain and range.
Absolute Value Functions
Graph y = |x| + 1. Identify the domain and range.
Create a table of values. x
|x| + 1 –3 4 –2 3 –1 2 1
Example 4

15. Graph the points and connect them.
Absolute Value Functions
Graph the points and connect them.
Answer:
The domain is all real numbers. The range is {y | y ≥ 1}.
Example 4

16. Identify the function shown by the graph.
A. y = |x| – 1
B. y = |x – 1| – 1
C. y = |x – 1|
D. y = |x + 1| – 1
Example 4