Define piecewise function, use graph of piecewise function and function to evaluate, and verifying an equation matches graph.

1. Piecewise Functions
What are they?
Evaluating from a Graph
Evaluating from a Function
Verify Function is the Graph

2. What is a Piecewise Function?

3. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.

4. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.

5. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?

6. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!

7. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!
• A piecewise function will have the same
number of equations as the graph has
pieces.
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

8. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!
• A piecewise function will have the same
number of equations as the graph has
pieces.
• Each equation in the function represents a
piece on the graph. f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

9. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!
• A piecewise function will have the same
number of equations as the graph has
pieces.
• Each equation in the function represents a
piece on the graph. f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

10. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!
• A piecewise function will have the same
number of equations as the graph has
pieces.
• Each equation in the function represents a
piece on the graph. f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

11. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!
• A piecewise function will have the same
number of equations as the graph has
pieces.
• Each equation in the function represents a
piece on the graph. f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

12. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!
• A piecewise function will have the same
number of equations as the graph has
pieces.
• Each equation in the function represents a
piece on the graph. f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

13. What is a Piecewise Function?
• A piecewise function is exactly what it
sounds like. It is a function made of
pieces.
• Look at the graph to the right. Notice if
you apply the vertical line test it passes.
Thus, it is a function.
• The pieces look very different so how can
a function model the graph?
• This is where piecewise functions come in!
• A piecewise function will have the same
number of equations as the graph has
pieces.
• Each equation in the function represents a
piece on the graph.
• Piecewise function always have domain
restrictions so you know which equation
matches which piece.
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

14. Evaluating From a Graph

15. • If you need to know a value
but are only given the graph,
how do find the value?
Evaluating From a Graph

16. • If you need to know a value
but are only given the graph,
how do find the value?
• Go to the x value on the x-
axis.
Evaluating From a Graph

17. • If you need to know a value
but are only given the graph,
how do find the value?
• Go to the x value on the x-
axis.
• Move vertically until you hit
the function.
Evaluating From a Graph

18. • If you need to know a value
but are only given the graph,
how do find the value?
• Go to the x value on the x-
axis.
• Move vertically until you hit
the function.
• Find the point.
Evaluating From a Graph

19. • If you need to know a value
but are only given the graph,
how do find the value?
• Go to the x value on the x-
axis.
• Move vertically until you hit
the function.
• Find the point.
• The y-coordinate is the value
of the function at that x.
Evaluating From a Graph

20. • Find f(-1)
Evaluating From a Graph (continued)

21. • Find f(-1)
• Move horizontally on the x-axis
to find x = -1.
Evaluating From a Graph (continued)

22. • Find f(-1)
• Move horizontally on the x-axis
to find x = -1.
• Move vertically until you hit the
function.
Evaluating From a Graph (continued)

23. • Find f(-1)
• Move horizontally on the x-axis
to find x = -1.
• Move vertically until you hit the
function.
• In this case, the function is an
end point. Because it is a closed circle,
the point is part of the function. So the point is (-1, 1).
Evaluating From a Graph (continued)

24. • Find f(-1)
• Move horizontally on the x-axis
to find x = -1.
• Move vertically until you hit the
function.
• In this case, the function is an
end point. Because it is a closed circle,
the point is part of the function. So the point is (-1, 1).
• The value of the function is the y-coordinate of the point
on the function.
Evaluating From a Graph (continued)

25. • Find f(-1)
• Move horizontally on the x-axis
to find x = -1.
• Move vertically until you hit the
function.
• In this case, the function is an
end point. Because it is a closed circle,
the point is part of the function. So the point is (-1, 1).
• The value of the function is the y-coordinate of the point
on the function.
• Thus, f(-1) = 1.
Evaluating From a Graph (continued)

26. • f(-5)
• f(-2)
• f(0)
• f(2)
• Check your answers on the next slide.
Try Some
Use the graph to evaluate each of the following.
Check your answers on the next slide.

27. • f(-5) = 2
• f(-2) = undefined
• f(0) = 1
• f(2) = 4
Check
How did you do? Check your answers below and
then check out this video to see how each is found.

28. • Evaluate f(-2).
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

29. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

30. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

31. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
• The right column in the
piecewise function is the
domain for each equation.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

32. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
• The right column in the
piecewise function is the
domain for each equation.
• Find the domain in which the
x falls and use that equation
to evaluate the function.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

33. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
• The right column in the
piecewise function is the
domain for each equation.
• Find the domain in which the
x falls and use that equation
to evaluate the function.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$
• Notice -2 falls in the first domain
because it is less than 0.

34. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
• The right column in the
piecewise function is the
domain for each equation.
• Find the domain in which the
x falls and use that equation
to evaluate the function.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$
• Notice -2 falls in the first domain
because it is less than 0.
• Thus, use the first equation to
evaluate.

35. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
• The right column in the
piecewise function is the
domain for each equation.
• Find the domain in which the
x falls and use that equation
to evaluate the function.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$
• Notice -2 falls in the first domain
because it is less than 0.
• Thus, use the first equation to
evaluate.
f x( )= 2x + 3

36. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
• The right column in the
piecewise function is the
domain for each equation.
• Find the domain in which the
x falls and use that equation
to evaluate the function.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$
• Notice -2 falls in the first domain
because it is less than 0.
• Thus, use the first equation to
evaluate.
f −2( )= 2 −2( )+ 3
= −4 + 3
= −1
f x( )= 2x + 3

37. • Evaluate f(-2).
• When asked to evaluate from
a function, how do you know
which equation to use?
• Remember the number inside
the parentheses for f(-2) is the
x.
• The right column in the
piecewise function is the
domain for each equation.
• Find the domain in which the
x falls and use that equation
to evaluate the function.
Evaluating From Function
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$
• Notice -2 falls in the first domain
because it is less than 0.
• Thus, use the first equation to
evaluate.
• Therefore, f(-2) = -1.
f −2( )= 2 −2( )+ 3
= −4 + 3
= −1
f x( )= 2x + 3

38. • Evaluate f(0).
Evaluating From Equation (continued)
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

39. • Evaluate f(0).
• This one is tricky. Notice the first domain is less
than 0 and the second domain is greater than or
equal to 1. Because we need to evaluate when x
is 0, no function exists for this domain.
Evaluating From Equation (continued)
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

40. • Evaluate f(0).
• This one is tricky. Notice the first domain is less
than 0 and the second domain is greater than or
equal to 1. Because we need to evaluate when x
is 0, no function exists for this domain.
• Thus, this function is undefined when x is 0. So,
f(0) = undefined.
Evaluating From Equation (continued)
f x( )=
2x+ 3 x < 0
2x
x ≥1
"
#
$
%$

41. • Evaluate f(8).
• Evaluate f(-5).
• Evaluate f(0).
• Evaluate f(-4).
• Evaluate f(3)
• Evaluate f(5).
Try Some
Use the piecewise function to evaluate each. Check
your answers on the next slide.
f x( )=
3x2
x ≤ −4
11 −4 < x ≤ 3
x+2 x > 5
#
$
%
&
%

42. • f(8) = 10
• f(-5) = 75
• f(0) = 11
• f(-4) = 48
• f(3) = 11
• f(5) = undefined
Check
How did you do? Check your answers below and then
check out this video to see how each is evaluated
f x( )=
3x2
x ≤ −4
11 −4 < x ≤ 3
x+2 x > 5
#
$
%
&
%

43. Verifying Piecewise from Graph
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

44. • You may need to verify a piecewise
function. Such as on a multiple choice
question. Do this by testing end points
to confirm/eliminate answer choices.
Keep doing this until you have only 1
answer choice left.
Verifying Piecewise from Graph
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

45. • You may need to verify a piecewise
function. Such as on a multiple choice
question. Do this by testing end points
to confirm/eliminate answer choices.
Keep doing this until you have only 1
answer choice left.
• Such as check x = -8 by substituting
into the first function, x + 5. Remember
that y can be substituted for f(x).
Verifying Piecewise from Graph
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

46. • You may need to verify a piecewise
function. Such as on a multiple choice
question. Do this by testing end points
to confirm/eliminate answer choices.
Keep doing this until you have only 1
answer choice left.
• Such as check x = -8 by substituting
into the first function, x + 5. Remember
that y can be substituted for f(x).
Verifying Piecewise from Graph
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&
y = x +5
y = −8+5
y = −3

47. • You may need to verify a piecewise
function. Such as on a multiple choice
question. Do this by testing end points
to confirm/eliminate answer choices.
Keep doing this until you have only 1
answer choice left.
• Such as check x = -8 by substituting
into the first function, x + 5. Remember
that y can be substituted for f(x).
• Thus, (-8, -3) should be
a point on the graph and it is.
Verifying Piecewise from Graph
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&
y = x +5
y = −8+5
y = −3

48. Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

49. • For the end points where they
should not be equal, make sure
the graph shows and open circle.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

50. • For the end points where they
should not be equal, make sure
the graph shows and open circle.
• Such as look at the third piece
where the domain is less than 3.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

51. • For the end points where they
should not be equal, make sure
the graph shows and open circle.
• Such as look at the third piece
where the domain is less than 3.
• Substitute x = 3 into the equation.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

52. • For the end points where they
should not be equal, make sure
the graph shows and open circle.
• Such as look at the third piece
where the domain is less than 3.
• Substitute x = 3 into the equation.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&
y = x2
y = 32
y = 9

53. • For the end points where they
should not be equal, make sure
the graph shows and open circle.
• Such as look at the third piece
where the domain is less than 3.
• Substitute x = 3 into the equation.
• Thus, the point (3, 9) should
be an open circle.
Look at the graph and notice it is.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&
y = x2
y = 32
y = 9

54. • What if after evaluating all end
points you have answer choices
left?
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

55. • What if after evaluating all end
points you have answer choices
left?
• Pick a point in the domain to verify
the equation works.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

56. • What if after evaluating all end
points you have answer choices
left?
• Pick a point in the domain to verify
the equation works.
• Let’s look at when x = 10. Notice it
falls in the last domain. Thus,
evaluate x = 10 in the last equation.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&

57. • What if after evaluating all end
points you have answer choices
left?
• Pick a point in the domain to verify
the equation works.
• Let’s look at when x = 10. Notice it
falls in the last domain. Thus,
evaluate x = 10 in the last equation.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&
y = log x
y = log10
y =1

58. • What if after evaluating all end
points you have answer choices
left?
• Pick a point in the domain to verify
the equation works.
• Let’s look at when x = 10. Notice it
falls in the last domain. Thus,
evaluate x = 10 in the last equation.
• Thus, the point (10, 1) should
be on the last piece and notice it is.
Verifying Piecewise from Graph (continued)
f x( )=
x+5 x ≤ −8
sin x+ 3 −8 < x < −2
x2
−1≤ x < 3
log x x ≥ 3
$
%
&
&
'
&
&
y = log x
y = log10
y =1

59. • Piecewise functions are several different functions
grouped for specific domains.
Summary

60. • Piecewise functions are several different functions
grouped for specific domains.
• Evaluate on a graph by finding the x on the x-axis,
move vertically until you hit the function, write the
point, and the y-coordinate is the value of the function
at x.
Summary

61. • Piecewise functions are several different functions
grouped for specific domains.
• Evaluate on a graph by finding the x on the x-axis,
move vertically until you hit the function, write the
point, and the y-coordinate is the value of the function
at x.
• Evaluate equations by finding the domain in which the
x falls. Use x in the equation for that domain.
Summary

62. • Piecewise functions are several different functions
grouped for specific domains.
• Evaluate on a graph by finding the x on the x-axis,
move vertically until you hit the function, write the
point, and the y-coordinate is the value of the function
at x.
• Evaluate equations by finding the domain in which the
x falls. Use x in the equation for that domain.
• If x does not fall in the domain or hit the function on the
graph, the function is not defined at that value and the
function is said to be undefined at that value.
Summary