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nycguitarguy

Member

Registered: 2024-02-24

Posts: 549

Domain & Range of Piecewise Functions

Find the domain and range of the piecewise function below.

f(x) = 2 - x  if -3<= x < 1....top portion

f(x) = sqrt{x}  if x > 1....bottom portion

Bob

Administrator

Registered: 2010-06-20

Posts: 10,198

Re: Domain & Range of Piecewise Functions

Your latest 3 posts are all of a kind.  Up until now a single function equation has given you all you need to determine how the function behaves.

Now we have definitions in sections with different things happening according to a multiple set of rules.

But the method is still the same.  If you can sketch the graph correctly the answers will be easy to find.

A piecewise definition means work out the graph for some bits, then for the other bits. Maybe the two bits will join up, but that doesn't have to happen.

Bob
.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 549

Re: Domain & Range of Piecewise Functions

Your latest 3 posts are all of a kind.  Up until now a single function equation has given you all you need to determine how the function behaves.

Now we have definitions in sections with different things happening according to a multiple set of rules.

But the method is still the same.  If you can sketch the graph correctly the answers will be easy to find.

A piecewise definition means work out the graph for some bits, then for the other bits. Maybe the two bits will join up, but that doesn't have to happen.

Bob
.

Can you please graph this piecewise function using different colors for each part?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,198

Re: Domain & Range of Piecewise Functions

Sorry,  no access to my colored grapher at the moment. Please wait a few days.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 549

Re: Domain & Range of Piecewise Functions

Sorry,  no access to my colored grapher at the moment. Please wait a few days.

Bob

Ok. Thanks. I will wait for you to upload the graph.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,198

Re: Domain & Range of Piecewise Functions

hi

Back with my laptop now so I can do this.

The first part of the definition is a straight line so I worked out the end points ie when x = -3 and x = 1

Then I joined them with a line in red.

At this stage I was aware that the right hand endpoint is not included but I left the point in for now More on this after I do the second function.**

I then did all of f(x) = sqroot(x) and rubbed out the bit from (0,0) to (1,1). This part is in blue.

** Sometimes when a function is defined in parts, there is a discontinuity between the parts.  But, not so here. (1,1) is on both so it didn't matter that I left in the point.  The reason x=1 is not part of the first definition is you cannot have two separate definitions for the y value here even though both definitions would result in the same y (=1).

The blue curve will  continue for all x where x > 1.  Although the curve slopes less and less as x gets bigger, it is nevertheless increasing with no top limit. So bear that in mind when deciding the range.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 549

Re: Domain & Range of Piecewise Functions

hi

Back with my laptop now so I can do this.

The first part of the definition is a straight line so I worked out the end points ie when x = -3 and x = 1

Then I joined them with a line in red.

At this stage I was aware that the right hand endpoint is not included but I left the point in for now More on this after I do the second function.**

I then did all of f(x) = sqroot(x) and rubbed out the bit from (0,0) to (1,1). This part is in blue.

https://i.imgur.com/PoEC3JX.gif

** Sometimes when a function is defined in parts, there is a discontinuity between the parts.  But, not so here. (1,1) is on both so it didn't matter that I left in the point.  The reason x=1 is not part of the first definition is you cannot have two separate definitions for the y value here even though both definitions would result in the same y (=1).

The blue curve will  continue for all x where x > 1.  Although the curve slopes less and less as x gets bigger, it is nevertheless increasing with no top limit. So bear that in mind when deciding the range.

Bob

Part 1: f(x) = 2 - x for -3 ≤ x < 1
Domain: The domain of this part of the function is the set of all real numbers x such that -3 ≤ x < 1.
Range: The range of this part of the function is the set of all real numbers y such that y = 2 - x, where -3 ≤ x < 1.
The range can be found by considering the minimum and maximum values of the function in this interval.
The minimum value occurs when x = -3, giving f(x) = 2 - (-3) = 5.
The maximum value occurs when x = 0, giving f(x) = 2 - 0 = 2.
Therefore, the range of this part of the function is [2, 5].

You say?

Part 2: f(x) = √x for x > 1
Domain: The domain of this part of the function is the set of all real numbers x such that x > 1.
Range: The range of this part of the function is the set of all real numbers y such that y = √x, where x > 1.
The range can be found by considering the minimum and maximum values of the function in this interval.
The minimum value occurs when x = 1, giving f(x) = √1 = 1.
The maximum value is not bounded, as the function approaches positive infinity as x approaches positive infinity.
Therefore, the range of this part of the function is [1, ∞).

I say, the domain of the entire piecewise function is the set of all real numbers x such that -3 ≤ x < ∞, and the range of the entire piecewise function is [1, ∞).

You say?

P. S. You forgot to change my username.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,198

Re: Domain & Range of Piecewise Functions

Domain and range correct.

I've made a username suggestion in the other post.  I didn't want to try this from my phone in case I clicked the wrong thing.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 549

Re: Domain & Range of Piecewise Functions

Domain and range correct.

I've made a username suggestion in the other post.  I didn't want to try this from my phone in case I clicked the wrong thing.

Bob

Great. Cool. I need to work more on piecewise functions.