IB AA · 3A Concepts of a Function
f⁻¹(k) is the input that makes the output equal k.
Meaning
Do not start by building a full inverse formula unless you need it. Set f(x) = k and solve.
Domain
If the domain was restricted, keep only the solution that lies in the domain.
A table
Read backwards: find the output, then report its input.
A model
If f turns one unit into another, f⁻¹ turns it back.
A linear rule.
f⁻¹(10) is the input that makes f(x) = 10.
2x − 6 = 10
2x = 16
x = 8
f⁻¹(10) = 8
Answer: 8
f⁻¹(11) is the input that makes f(x) = 11.
3x − 1 = 11
3x = 12
x = 4
f⁻¹(11) = 4
Answer: 4
f⁻¹(17) is the input that makes f(x) = 17.
2x + 5 = 17
2x = 12
x = 6
f⁻¹(17) = 6
Answer: 6
f⁻¹(3) is the input that makes f(x) = 3.
−x + 8 = 3
−x = −5
x = 5
f⁻¹(3) = 5
Answer: 5
f⁻¹(20) is the input that makes f(x) = 20.
5x = 20
5x = 20
x = 4
f⁻¹(20) = 4
Answer: 4
f⁻¹(9) is the input that makes f(x) = 9.
4x − 7 = 9
4x = 16
x = 4
f⁻¹(9) = 4
Answer: 4
A restricted quadratic, a table, or a conversion.
f⁻¹(5) is the allowed input that makes f(x) = 5.
x² − 6x + 5 = 5
x² − 6x = 0
x(x − 6) = 0
x = 0 or x = 6
The domain says x ≤ 3, so x = 0.
f⁻¹(5) = 0
Answer: 0
f⁻¹(2) is the input whose output is 2.
f(1) = 2, so f⁻¹(2) = 1.
f(2) = 8, so the solution of f(x) = 8 is x = 2.
Answer: f⁻¹(2) = 1, x = 2
f(x) = 54x
f multiplies euros by 1.25 to get dollars.
f⁻¹(x) = 45x
f⁻¹ turns a dollar amount back into euros.
Answer: f(x) = 54x; f⁻¹ turns dollars into euros
g⁻¹(5) is the allowed input that makes g(x) = 5.
= 5
1 + x² = 25
x² = 24
x = or x = −
x ≤ 0, so x = −2
g⁻¹(5) = −2
Answer: −2
f⁻¹(1) is the input that makes f(x) = 1.
−2x + 9 = 1
−2x = −8
x = 4
f⁻¹(1) = 4
Answer: 4
The inside must be positive: 4x − 8 > 0.
x > 2
n(4) = log2 8
n(4) = 3
n⁻¹(2) is the input that makes n(x) = 2.
log2 4x − 8 = 2
4x − 8 = 4
x = 3
Answer: x > 2, n(4) = 3, n⁻¹(2) = 3
A square root, a log, and a model.
g⁻¹(5) is the allowed input that makes g(x) = 5.
= 5
1 + x² = 25
x² = 24
x = or x = −
x ≤ 0, so x = −2
g⁻¹(5) = −2
Answer: −2
The inside must be positive: 4x − 8 > 0.
x > 2
n(4) = log2 8
n(4) = 3
n⁻¹(2) is the input that makes n(x) = 2.
log2 4x − 8 = 2
4x − 8 = 4
x = 3
Answer: x > 2, n(4) = 3, n⁻¹(2) = 3
v(t) = 4t
v(3) = 4(3)
v(3) = 12
After 40 seconds the model gives 160 m/s, which is far faster than this cart can go.
The model is only reasonable for a short time.
Answer: 12 m/s; the model grows without a real speed limit
f⁻¹(6) is the input that makes f(x) = 6.
13x + 2 = 6
13x = 4
x = 12
f⁻¹(6) = 12
Answer: 12
f⁻¹(5) is the allowed input that makes f(x) = 5.
x² − 6x + 5 = 5
x² − 6x = 0
x(x − 6) = 0
x = 0 or x = 6
The domain says x ≤ 3, so x = 0.
f⁻¹(5) = 0
Answer: 0
f⁻¹(2) is the input whose output is 2.
f(1) = 2, so f⁻¹(2) = 1.
f(2) = 8, so the solution of f(x) = 8 is x = 2.
Answer: f⁻¹(2) = 1, x = 2
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