PreCalc 2 · Review
The domain is every input that is allowed. For a fraction, the bottom cannot be zero. For a square root, the inside cannot be negative.
What domain means
Domain is the list of inputs that work.
You cannot divide by zero.
You cannot take the square root of a negative number.
How to write it
For a fraction: set the denominator equal to zero and kick those numbers out.
Set-builder looks like this:
{x | x is a real number and x ≠ 3}
Read it: “all real x except 3.”
Square roots
Set the inside of the square root ≥ 0, then solve the inequality.
{x | x is a real number and x ≥ 3} means 3 and every number bigger than 3.
Find the domain of 1x − 4.
You cannot divide by zero.
x − 4 ≠ 0
x ≠ 4
{x | x is a real number and x ≠ 4}
Answer: {x | x is a real number and x ≠ 4}
Find the domain of 1x + 5.
You cannot divide by zero.
x + 5 ≠ 0
x ≠ −5
{x | x is a real number and x ≠ −5}
Answer: {x | x is a real number and x ≠ −5}
Find the domain of 3x.
You cannot divide by zero.
x ≠ 0
{x | x is a real number and x ≠ 0}
Answer: {x | x is a real number and x ≠ 0}
Find the domain of x + 2x − 1.
The bottom cannot be zero.
x − 1 ≠ 0
x ≠ 1
{x | x is a real number and x ≠ 1}
Answer: {x | x is a real number and x ≠ 1}
Find the domain of 72x + 8.
2x + 8 ≠ 0
2x ≠ −8
x ≠ −4
{x | x is a real number and x ≠ −4}
Answer: {x | x is a real number and x ≠ −4}
Find the domain of x − 3x + 6.
x + 6 ≠ 0
x ≠ −6
{x | x is a real number and x ≠ −6}
Answer: {x | x is a real number and x ≠ −6}
Find the domain of √(x − 3).
You cannot take the square root of a negative number.
x − 3 ≥ 0
x ≥ 3
{x | x is a real number and x ≥ 3}
Answer: {x | x is a real number and x ≥ 3}
Find the domain of √(x + 7).
You cannot take the square root of a negative number.
x + 7 ≥ 0
x ≥ −7
{x | x is a real number and x ≥ −7}
Answer: {x | x is a real number and x ≥ −7}
Find the domain of √x.
You cannot take the square root of a negative number.
x ≥ 0
{x | x is a real number and x ≥ 0}
Answer: {x | x is a real number and x ≥ 0}
Find the domain of x + 1x² − 5x + 6.
The bottom cannot be zero.
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x − 2 = 0 or x − 3 = 0
x ≠ 2, x ≠ 3
{x | x is a real number and x ≠ 2, x ≠ 3}
Answer: {x | x is a real number and x ≠ 2, x ≠ 3}
Write the domain of k(s) = s² + 42s² − s − 45 in set-builder notation.
The bottom cannot be zero.
2s² − s − 45 = 0
(2s + 9)(s − 5) = 0
2s + 9 = 0 or s − 5 = 0
s = −92 or s = 5
The top s² + 4 is never zero, so it does not kick out extra values.
{s | s is a real number and s ≠ −92, s ≠ 5}
Answer: {s | s is a real number and s ≠ −92, s ≠ 5}
Find the domain of x − 4x² − 9.
x² − 9 = 0
(x − 3)(x + 3) = 0
x ≠ 3, x ≠ −3
{x | x is a real number and x ≠ −3, x ≠ 3}
Answer: {x | x is a real number and x ≠ −3, x ≠ 3}
Find the domain of 2xx² + x − 6.
x² + x − 6 = 0
(x + 3)(x − 2) = 0
x ≠ −3, x ≠ 2
{x | x is a real number and x ≠ −3, x ≠ 2}
Answer: {x | x is a real number and x ≠ −3, x ≠ 2}
Find the domain of 5x² − 7x + 12.
x² − 7x + 12 = 0
(x − 3)(x − 4) = 0
x ≠ 3, x ≠ 4
{x | x is a real number and x ≠ 3, x ≠ 4}
Answer: {x | x is a real number and x ≠ 3, x ≠ 4}
Find the domain of x² + 1x² − x − 20.
x² − x − 20 = 0
(x − 5)(x + 4) = 0
x ≠ 5, x ≠ −4
{x | x is a real number and x ≠ −4, x ≠ 5}
Answer: {x | x is a real number and x ≠ −4, x ≠ 5}
Find the domain of √(2x − 6).
You cannot take the square root of a negative number.
2x − 6 ≥ 0
2x ≥ 6
x ≥ 3
{x | x is a real number and x ≥ 3}
Answer: {x | x is a real number and x ≥ 3}
Find the domain of √(4 − x).
You cannot take the square root of a negative number.
4 − x ≥ 0
4 ≥ x
x ≤ 4
{x | x is a real number and x ≤ 4}
Answer: {x | x is a real number and x ≤ 4}
Find the domain of √(3x + 9).
You cannot take the square root of a negative number.
3x + 9 ≥ 0
3x ≥ −9
x ≥ −3
{x | x is a real number and x ≥ −3}
Answer: {x | x is a real number and x ≥ −3}
Find the domain of x − 2x² − 4.
x² − 4 = 0
(x − 2)(x + 2) = 0
x ≠ 2, x ≠ −2
Even though (x − 2) cancels, x = 2 still makes the original bottom zero, so it is not allowed.
{x | x is a real number and x ≠ −2, x ≠ 2}
Answer: {x | x is a real number and x ≠ −2, x ≠ 2}
Find the domain of x + 3x² − 9.
x² − 9 = 0
(x − 3)(x + 3) = 0
x ≠ 3, x ≠ −3
x = −3 still makes the original bottom zero, so leave it out even if it cancels.
{x | x is a real number and x ≠ −3, x ≠ 3}
Answer: {x | x is a real number and x ≠ −3, x ≠ 3}
Find the domain of xx² − x.
x² − x = 0
x(x − 1) = 0
x ≠ 0, x ≠ 1
{x | x is a real number and x ≠ 0, x ≠ 1}
Answer: {x | x is a real number and x ≠ 0, x ≠ 1}
Find the domain of 2x − 6x² − 5x + 6.
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x ≠ 2, x ≠ 3
{x | x is a real number and x ≠ 2, x ≠ 3}
Answer: {x | x is a real number and x ≠ 2, x ≠ 3}
Find the domain of 1x³ − 4x.
x³ − 4x = 0
x(x² − 4) = 0
x(x − 2)(x + 2) = 0
x ≠ 0, x ≠ 2, x ≠ −2
{x | x is a real number and x ≠ −2, x ≠ 0, x ≠ 2}
Answer: {x | x is a real number and x ≠ −2, x ≠ 0, x ≠ 2}
Find the domain of x² + xx² − 1.
x² − 1 = 0
(x − 1)(x + 1) = 0
x ≠ 1, x ≠ −1
{x | x is a real number and x ≠ −1, x ≠ 1}
Answer: {x | x is a real number and x ≠ −1, x ≠ 1}
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