Functions
Inverse functions, logarithmic and exponential
Inverse Functions
The inverse of a function reverses the operation. If f(x) = 2x + 3, then f⁻¹(x) = (x − 3)/2. Graphically, the inverse is a reflection in the line y = x. A function must be one-to-one to have an inverse that is also a function.
Example
Finding Inverses
f(x) = 3x − 6
Swap x and y: x = 3y − 6
Solve for y: y = (x + 6)/3
f⁻¹(x) = (x + 6)/3
f(x) = x² (restrict to x ≥ 0)
f⁻¹(x) = √x
Logarithmic and Exponential Functions
If f(x) = aˣ, then f⁻¹(x) = log_a(x).
y = 2ˣ and y = log₂(x) are inverses of each other.
The exponential graph passes through (0,1) with asymptote y = 0.
The log graph passes through (1,0) with asymptote x = 0.
Note
Remember
To find an inverse: swap x and y, then solve for y. The graph of f⁻¹ is the reflection of f in y = x. Restrict the domain of f(x) = x² to ensure the inverse is a function.
Key Vocabulary
Inverse functionA function that reverses the operation of another function
One-to-oneEach output comes from exactly one input
Exponential functionA function of the form y = aˣ
LogarithmThe inverse of an exponential: if aˣ = b then x = log_a(b)
AsymptoteA line that a graph approaches but never touches
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Inverse function
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