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• piecewise-defined function
• piecewise-linear function
• step function
• greatest integer function
• absolute value function
A function that is written using two or more
expressions is called a piecewise function.
Piecewise-Defined Function
Step 1 Graph the linear function f(x) = x –1 for x ≤ 3.
Begin with
f(3) = 3 -1=2
Mark a closed circle at (3, 2)
Use the slope 1/1 to graph the
rest of the function (left).
Piecewise-Defined Function
Step 2 Graph the constantfunction f(x) = –1 forx > 3.
Begin with f(3) = -1
an open circle at(3, –1)
and draw a horizontal ray to the
right.
Piecewise-Defined Function
Answer: The function is defined for all values of x, so the domain is all realnumbers. The values that arey-coordinates of points on thegraph are all real numbersless than or equal to 2, so therange is {y | y ≤ 2}.
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. domain: all real numbersrange: all real numbers
B. domain: all real numbersrange: {y|y > –1}
C. domain: all real numbersrange: {y|y > –1 or y = –3}
D. domain: {x|x > –1 or x = –3}range: all real numbers
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}.
Write a Piecewise-Defined Function
Write the piecewise-defined function.
Answer:
A. A
B. B
C. C
D. D A B C D
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Identify the piecewise-defined function shown in the graph.
A.
B.
C.
D.
A graph of step function consists of a line
segments.
The greatest integer function written
f(x)=[x], is a one kind of a step function.
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.
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.
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.
A. A
B. B
C. C
D. D A B C D
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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.
An absolute value function is a function that
contains an algebraic expression within
absolute value symbols.
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
0 1
1 2
2 3
3 4
Absolute Value Functions
Graph the points and connect them.
Answer:
The domain is all realnumbers. The range is {y | y ≥ 1}.
A. A
B. B
C. C
D. D A B C D
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A. y = |x| – 1
B. y = |x – 1| – 1
C. y = |x – 1|
D. y = |x + 1| – 1
Identify the function shown by the graph.
