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**nycguitarguy****Member**- Registered: 2024-02-24
- Posts: 398

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

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,109

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

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**nycguitarguy****Member**- Registered: 2024-02-24
- Posts: 398

Bob wrote:

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?

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,109

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

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**nycguitarguy****Member**- Registered: 2024-02-24
- Posts: 398

Bob wrote:

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.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,109

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

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**nycguitarguy****Member**- Registered: 2024-02-24
- Posts: 398

Bob wrote:

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.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,109

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

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

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**nycguitarguy****Member**- Registered: 2024-02-24
- Posts: 398

Bob wrote:

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.

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