presentation about the piecewise functions

2. Definition:Definition:
Piecewise FunctionPiecewise Function –a–a
function defined by two orfunction defined by two or
more functions over amore functions over a
specified domain.specified domain.

3. What do they look like?
f(x) =
x2
+ 1 , x < 0
x – 1 , x ≥ 0
You can EVALUATE piecewise
functions.
You can GRAPH piecewise functions.

4. Evaluating Piecewise Functions:
Evaluating piecewise functions is just
like evaluating functions that you are
already familiar with.
f(x) =
x2
+ 1 , x < 0
x – 1 , x ≥ 0
Let’s calculate f(2).
You are being asked to find y when
x = 2. Since 2 is ≥ 0, you will only
substitute into the second part of
the function.
f(2) = 2 – 1 = 1

5. f(x) =
x2
+ 1 , x < 0
x – 1 , x ≥ 0
Let’s calculate f(-2).
You are being asked to find y when
x = -2. Since -2 is < 0, you will only
substitute into the first part of
the function.
f(-2) = (-2)2
+ 1 = 5

6. Your turn:
f(x) =
2x + 1, x < 0
2x + 2, x ≥ 0
Evaluate the following:
f(-2) = -3?
f(0) = 2?
f(5) = 12?
f(1) = 4?

7. One more:
f(x) =
3x - 2, x < -2
-x , -2 ≤ x < 1
x2
– 7x, x ≥ 1
Evaluate the following:
f(-2) = 2?
f(-4) = -14?
f(3) = -12?
f(1) = -6?

8. Graphing Piecewise Functions:
f(x) =
x2
+ 1 , x < 0
x – 1 , x ≥ 0
Determine the shapes of the graphs.
Parabola and Line
Determine the boundaries of each graph.
Graph the
parabola where x
is less than zero. •
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Graph the line
where x is
greater than or
equal to zero. •
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3x + 2, x < -2
-x , -2 ≤ x < 1
x2
– 2, x ≥ 1
f(x) =
Graphing Piecewise Functions:
Determine the shapes of the graphs.
Line, Line, Parabola
Determine the boundaries of each graph.
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10. Graphing Piecewise Functions
( )
x 4 x 4
2x
x 3 x
1
1
g x 5 4 x+ − ≤
− + >
 + < −
= ≤




Domain - ( ),−∞ ∞
Range - ( ), 7−∞

11. ( )
3 7 x 4
1
x 2 4 x 0
2
1
x 4
x
0 x 5
5 x 7
g
−
− −
− <
− − <



= 
<
< ≤ −
+ ≤ ≤
≤



Domain - (-7, 7]
Range - (-4, -2), [-1, 4]

12. ( )
1
x 6 x 3
3
x 1 3 x 0h x
x 4 0 x 3
x 3 3 x 7

− − ≤ ≤ −
 + − < ≤= 
 + < <

− + ≤ ≤
Domain - [-6, 7]
Range - [-4, 2], (4, 7)

13. ( )
4 x 3
h x 2x 3 3 x 4
4x 7 x 6
≤ −

= + − < ≤
 − ≥

14. Piecewise Function – Domain and Range
Domain -
Range -
Domain - (-6, 7)
Range - [-1, 5 )
[-7, 7]
(-4.5,-1], [0, 4)

15. Domain -
Range -
Domain -
Range -
(-7, -1), (-1, 7]
[-1, 5), [6, 6]
(-7, 4), [5, 7)
[-7, -5), (-2, 7)

16. Domain -
Range -
Domain -
Range -
( ),−∞ ∞
( ], 4−∞
[-1, 5]
[-5, 3]