IB AA · 3A Concepts of a Function
Find every real input the rule allows.
Polynomials
A polynomial is defined for every real number.
Denominator
Set the denominator not equal to 0. A squared factor is still 0 at that input.
Square root
The inside must be ≥ 0. If the root is in a denominator, the inside must be > 0.
Logarithm
The inside must be positive. It cannot be 0.
One restriction.
f(x) =
The inside of a square root must be at least 0.
x + 5 ≥ 0
x ≥ −5
Answer: x ≥ −5
A polynomial uses only whole-number powers of x.
There is no denominator and no root to restrict x.
The domain is every real number.
Answer: all real numbers
f(x) = 1x − 4
The denominator cannot be 0.
(x − 4) ≠ 0
x ≠ 4
Answer: x ≠ 4
f(x) =
The inside of a square root must be at least 0.
x − 3 ≥ 0
x ≥ 3
Answer: x ≥ 3
f(x) = ln(x − 2)
The inside of a logarithm must be positive.
x − 2 > 0
x > 2
Answer: x > 2
f(x) = 1x + 1
The denominator cannot be 0.
(x + 1) ≠ 0
x ≠ −1
Answer: x ≠ −1
A negative coefficient, or two restrictions.
f(x) =
The inside of a square root must be at least 0.
3x − 9 ≥ 0
3x ≥ 9
x ≥ 3
Answer: x ≥ 3
f(x) =
The inside of a square root must be at least 0.
2x − 6 ≥ 0
2x ≥ 6
x ≥ 3
Answer: x ≥ 3
f(x) =
The inside of a square root must be at least 0.
−x + 5 ≥ 0
−x ≥ −5
Dividing by a negative number reverses the inequality.
x ≤ 5
Answer: x ≤ 5
f(x) = ln(−3x + 12)
The inside of a logarithm must be positive.
−3x + 12 > 0
−3x > −12
Dividing by a negative number reverses the inequality.
x < 4
Answer: x < 4
f(x) = 3(x − 5)²
The denominator cannot be 0.
(x − 5)² ≠ 0
x ≠ 5
Answer: x ≠ 5
The inside of the square root must be at least 0, and the denominator cannot be 0.
x − 2 ≥ 0, so x ≥ 2
x − 7 ≠ 0, so x ≠ 7
The domain is x ≥ 2 and x ≠ 7.
Answer: x ≥ 2, x ≠ 7
A square, a log of a quadratic, or a root in the denominator.
f(x) = ln(x)
The inside of a logarithm must be positive.
x > 0
x > 0
Answer: x > 0
The inside must be positive.
x² − 16 > 0
x² > 16
x < −4 or x > 4
Answer: x < −4 or x > 4
The inside of the square root must be positive, because it is also a denominator.
x − 4 > 0
x > 4
Answer: x > 4
f(x) = 4(x)²
The denominator cannot be 0.
(x)² ≠ 0
x ≠ 0
Answer: x ≠ 0
f(x) = log2 −2x + 10
The inside of a logarithm must be positive.
−2x + 10 > 0
−2x > −10
Dividing by a negative number reverses the inequality.
x < 5
Answer: x < 5
f(x) = ln(4x − 20)
The inside of a logarithm must be positive.
4x − 20 > 0
4x > 20
x > 5
Answer: x > 5
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