Worked examples
The question
Question: find the domain of f(x) = 1/(x² − 4).
f(x) = 1x2 − 4
The domain is all the x's we can put in.
domain = the x's that can go in
1 Find where the bottom is 0
We can never divide by 0.
So the bottom can never be 0.
÷ 0: not allowed
Which x's make x² − 4 zero?
There is an x², so there can be two.
We solve x² − 4 = 0 with the quadratic lesson.
x2 − 4 = 0
Careful!
x² = 4 has two answers: 2 and −2.
Both make the bottom 0, so both are not allowed.
Careful: not x ≠ 2 only ✗
2 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 4, we add 4 on both sides.
x2 − 4 + 4 = 0 + 4
0 + 4 = 4.
Now we have x² = 4.
x2 = 4
3 Square root of both sides
x squared is 4.
So x is the square root of 4, or minus it.
Minus times minus is plus, so the negative one works too.
x² = k → x = √k or x = −√k
2 × 2 = 4, (−2) × (−2) = 4
2 times 2 is 4.
So x = 2 or x = −2.
x = 2 or x = −2
4 Write the domain
Both are not allowed.
The domain is all real numbers except −2 and 2: x ≠ −2 and x ≠ 2.
domain : x ≠ −2 and x ≠ 2
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.
interval: (−∞, −2) ∪ (−2, 2) ∪ (2, ∞)
6 Check
Let's check.
Put 2 in: the bottom is 0.
Not allowed.
x = 2: 122 − 4 = 10 not allowed
Now 0: the bottom is −4, not 0.
It works.
x = 0: 102 − 4 = 1−4 = −14
Let's look back.
First: find where the bottom is 0.
Then: get x² alone.
Then: square root of both sides.
Then: write the domain.
Then: show it on a number line.
Last: check.
Answer domain : x ≠ −2 and x ≠ 2
▶ Watch it solved
The question
Question: find the domain of f(x) = 5/(x² − 25).
f(x) = 5x2 − 25
The domain is all the x's we can put in.
domain = the x's that can go in
1 Find where the bottom is 0
We can never divide by 0.
So the bottom can never be 0.
÷ 0: not allowed
Which x's make x² − 25 zero?
There is an x², so there can be two.
We solve x² − 25 = 0 with the quadratic lesson.
x2 − 25 = 0
Careful!
x² = 25 has two answers: 5 and −5.
Both make the bottom 0, so both are not allowed.
Careful: not x ≠ 5 only ✗
2 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 25, we add 25 on both sides.
x2 − 25 + 25 = 0 + 25
0 + 25 = 25.
Now we have x² = 25.
x2 = 25
3 Square root of both sides
x squared is 25.
So x is the square root of 25, or minus it.
Minus times minus is plus, so the negative one works too.
x² = k → x = √k or x = −√k
5 × 5 = 25, (−5) × (−5) = 25
5 times 5 is 25.
So x = 5 or x = −5.
x = 5 or x = −5
4 Write the domain
Both are not allowed.
The domain is all real numbers except −5 and 5: x ≠ −5 and x ≠ 5.
domain : x ≠ −5 and x ≠ 5
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.
interval: (−∞, −5) ∪ (−5, 5) ∪ (5, ∞)
6 Check
Let's check.
Put 5 in: the bottom is 0.
Not allowed.
x = 5: 552 − 25 = 50 not allowed
Now 0: the bottom is −25, not 0.
It works.
x = 0: 502 − 25 = 5−25 = −15
Let's look back.
First: find where the bottom is 0.
Then: get x² alone.
Then: square root of both sides.
Then: write the domain.
Then: show it on a number line.
Last: check.
Answer domain : x ≠ −5 and x ≠ 5
▶ Watch it solved
The question
Question: find the domain of f(x) = 2/(x² − 5x + 4).
f(x) = 2x2 − 5x + 4
The domain is all the x's we can put in.
domain = the x's that can go in
1 Find where the bottom is 0
We can never divide by 0.
So the bottom can never be 0.
÷ 0: not allowed
Which x's make x² − 5x + 4 zero?
There is an x², so there can be two.
We solve x² − 5x + 4 = 0 with the quadratic lesson.
x2 − 5x + 4 = 0
Careful!
The bottom is 0 when either bracket is 0.
So two x's are not allowed.
Careful: each bracket = 0 gives an x that is not allowed
2 Factor
We need two numbers.
They multiply to 4, and they add to −5.
? × ? = 4
? + ? = −5
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
× is + → same signs: both −
We list the pairs that make 4, smallest first.
We add each pair.
We need 5.
1 and 4 give 5.
Yes!
With the signs: −1 and −4.
1 × 4: 1 + 4 = 5 ✓
So it is (x − 1)(x − 4) = 0.
(x − 1)(x − 4) = 0
3 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
A × B = 0 → A = 0 or B = 0
We make each bracket 0.
Two small equations.
x − 1 = 0 or x − 4 = 0
In x − 1 = 0, we add 1 on both sides.
In x − 4 = 0, we add 4 on both sides.
x − 1 + 1 = 0 + 1 or x − 4 + 4 = 0 + 4
So x = 1 or x = 4.
x = 1 or x = 4
4 Write the domain
Both are not allowed.
The domain is all real numbers except 1 and 4: x ≠ 1 and x ≠ 4.
domain : x ≠ 1 and x ≠ 4
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.
interval: (−∞, 1) ∪ (1, 4) ∪ (4, ∞)
6 Check
Let's check.
Put 4 in: the bottom is 0.
Not allowed.
x = 4: 242 − 5(4) + 4 = 20 not allowed
Now 0: the bottom is 4, not 0.
It works.
x = 0: 202 − 5(0) + 4 = 24 = 12
Let's look back.
First: find where the bottom is 0.
Then: factor.
Then: one bracket is 0.
Then: write the domain.
Then: show it on a number line.
Last: check.
Answer domain : x ≠ 1 and x ≠ 4
▶ Watch it solved
