Math · College algebra · Concept
Composite and inverse functions
Composition applies one function to the output of another: (f ∘ g)(x) = f(g(x)), with g applied first. An inverse function undoes a function, so f⁻¹(f(x)) = x. Only a one-to-one function, one whose graph passes the horizontal line test, has an inverse; to find it, solve y = f(x) for x, then swap the names of the variables.
Composition works from the inside out
To evaluate f(g(x)), apply g first and feed its output to f. Order matters: f(g(x)) and g(f(x)) are usually different functions.
The domain of a composition
x must be in the domain of g, and g(x) must be in the domain of f. For f(x) = √x and g(x) = x − 4, f(g(x)) = √(x − 4) needs x ≥ 4.
Reading a function as a composition
Seeing √(x² + 1) as f(g(x)), with inner function g(x) = x² + 1 and outer function f(u) = √u, is the step the chain rule depends on.
Inverse functions undo each other
If f takes a to b, then f⁻¹ takes b back to a. So f⁻¹(f(x)) = x on the domain of f, and f(f⁻¹(x)) = x on the domain of f⁻¹. The notation f⁻¹ means the inverse function, not 1/f.
One-to-one functions and the horizontal line test
An inverse exists only if different inputs always give different outputs. On a graph, no horizontal line may cross it more than once. y = x² fails, because f(−2) = f(2) = 4, but it passes on the restricted domain x ≥ 0, where its inverse is √x.
Finding an inverse
Write y = f(x), solve for x, then swap the names x and y. The graph of f⁻¹ is the graph of f reflected across the line y = x, so the domain and the range trade places.
Common mistakes
- Reading f⁻¹(x) as 1/f(x): for f(x) = 3x − 4, f⁻¹(x) = (x + 4)/3, not 1/(3x − 4).
- Applying the functions in the wrong order: f(g(x)) applies g first.
- Multiplying instead of composing: f(g(x)) is not f(x)·g(x).
- Inverting a function that is not one-to-one without first restricting its domain.
- Forgetting that the domain of f(g(x)) must respect both functions.
Key terms
- Composition of functions
- Feeding one function’s output into another, written f(g(x)). The inner output has to be an allowed input for the outer function.
- Inverse function
- A function that undoes another: if f(2) = 5, then f⁻¹(5) = 2. It exists only when each output comes from one input, and f⁻¹ does not mean 1/f.
- One-to-one function
- A function in which different inputs always give different outputs. Its graph passes the horizontal line test: no horizontal line meets it more than once. Exactly the one-to-one functions have inverses.
- Domain
- The set of inputs for which a function or expression is defined. In the real numbers that rules out zero denominators, negative numbers under even roots and inputs of logarithms that aren’t positive.
- Function range
- All the output values a function actually takes. A graph’s viewing window can hide some of them.
Work through an example
For f(x) = 2x + 1 and g(x) = x², find (f ∘ g)(x) and (g ∘ f)(x), and evaluate both at x = 3.
Compose two functions in both orders →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Check the values in Math See both composites in Graph Open worked example on a board Composition in Math ReferenceYour existing work stays on this device. Examples open as editable copies.