Domain and range of (1, 4), (2, 5), (3, 4) Easy

The question

Question: find the domain and range of the points (1, 4), (2, 5), (3, 4).

The domain is all the x's we can put in.

The range is all the answers, the y's, that can come out.

1 Read the x's

The first number of each point is its x: 1, 2 and 3.

So the domain is {1, 2, 3}.

2 Read the y's

The second number of each point is its y: 4, 5 and 4.

4 is there twice.

We write it once.

So the range is {4, 5}.

Let's look back.

First: read the x's.

Last: read the y's.

▶ Watch it solved

Domain of f(x) = 5/(2x - 6) Medium

The question

Question: find the domain of f(x) = 5/(2x − 6).

The domain is all the x's we can put in.

1 Find where the bottom is 0

We can never divide by 0.

So the bottom of the fraction can never be 0.

Which x makes the bottom 0?

We solve 2x − 6 = 0.

We want x by itself.

Two things are stuck to x: times 2 and − 6.

The − 6 is on the outside, so it comes off first.

Then the times 2.

2 Remove the − 6

To remove − 6, we add 6.

We do it on both sides, so they stay equal.

− 6 and + 6 make zero.

Cross them out.

0 + 6 = 6.

Now we have 2x = 6.

3 Remove the 2

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2 divided by 2 is 1.

So just x is left.

6 divided by 2 is 3.

So x = 3!

4 Write the domain

x = 3 is the only x that is not allowed.

The domain is all real numbers except 3: x ≠ 3.

5 Show it on a number line

Books also write the answer like this.

It is called interval notation.

A round bracket means the end is not in, like the open circle.

∞ means it goes on for ever.

∪ means or.

6 Check

Let's check.

Put 3 in: the bottom is 0.

Dividing by 0 is not allowed.

Now 4: the bottom is 2, not 0.

It works.

Let's look back.

First: find where the bottom is 0.

Then: remove the − 6.

Then: remove the 2.

Then: write the domain.

Then: show it on a number line.

Last: check.

▶ Watch it solved

Domain of f(x) = √(6 - 2x) Harder

The question

Question: find the domain of f(x) = √(6 − 2x).

The domain is all the x's we can put in.

1 The inside is 0 or more

No real number times itself is negative.

3 times 3 is 9.

Minus 3 times minus 3 is 9 too.

So we can't take the square root of a negative number.

So the inside, 6 − 2x, must be 0 or more.

We want x by itself.

Two things are stuck to x: times −2 and + 6.

The + 6 is on the outside, so it comes off first.

Then the times −2.

2 Remove the 6

To remove 6, we take away 6.

We do it on both sides, so the sign stays true.

6 and − 6 make zero.

Cross them out.

0 − 6 = −6.

Now we have −2x ≥ −6.

3 Remove the −2

−2x means −2 times x.

We want just x.

To remove times −2, we divide by −2.

We do it on both sides.

We divide by a negative number.

That turns the sign around.

Greater than or equal to becomes less than or equal to.

−2 divided by −2 is 1.

So just x is left.

−6 divided by −2 is 3.

So x is less than or equal to 3.

Careful!

Many people forget to turn the sign.

They get x ≥ 3.

But we divided by a negative number.

These are the x's that can go in.

So the domain is x ≤ 3.

4 Show it on a number line

Books also write the answer like this.

It is called interval notation.

A square bracket means the end is in, like the filled circle.

∞ means it goes on for ever.

5 Check

Let's check.

Put 4 in, just outside.

The inside is −2: negative.

Not allowed.

Now 1, inside: the inside is 4, and the square root of 4 is 2.

It works.

Let's look back.

First: the inside is 0 or more.

Then: remove the 6.

Then: remove the −2.

Then: show it on a number line.

Last: check.

▶ Watch it solved