Functions: notation, inverse and composite

Algebra • Notation, vocabulary and manipulation

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What is the horizontal line test used for?

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The horizontal line test indicates whether a function is one-to-one by checking if any horizontal line meets the graph more than once; if so, the function is not one-to-one.

Key concepts

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Function notation and evaluation

A function appears as f(x) = expression. The symbol f names the rule and x denotes the input. Substituting a number into x causes the expression to evaluate and produce the corresponding output. For example, if f(x) = 2x + 3 then f(4) = 2(4) + 3 = 11. The domain specifies allowable inputs and the range lists possible outputs.

Interpreting expressions as functions

An algebraic expression such as 3x − 5 can be treated as a function where the input is x and the output is 3x − 5. Writing the expression in function form, g(x) = 3x − 5, clarifies the input-output relationship. The representation as a 'machine' explains that each input passes through the same fixed steps and yields a single output.

Inverse function: definition and existence

An inverse function reverses the effect of the original function so that f^{-1}(f(x)) = x for all x in the original domain. An inverse exists only when each output of the original function comes from exactly one input (the function is one-to-one). Non-one-to-one functions require restriction of domain before an inverse exists.

Finding the inverse algebraically

To find an inverse, write y = f(x), swap x and y, then solve for y. The resulting expression is f^{-1}(x). For example, for f(x) = 2x + 3 set y = 2x + 3, swap to x = 2y + 3, solve to get y = (x − 3)/2, so f^{-1}(x) = (x − 3)/2. The swap enforces the reversal of input and output roles.

Composite functions and order

A composite function applies one function after another. The notation f(g(x)) means apply g to x, then apply f to the result. Order matters because f(g(x)) normally differs from g(f(x)). Evaluation of a composite follows the inner-then-outer sequence, so substitution of g(x) into f occurs directly in the expression for f.

Verifying inverses and domain effects

Verification uses composition: f(f^{-1}(x)) and f^{-1}(f(x)) should both simplify to x on the appropriate domains. Domain restrictions in either function transfer into the composite or inverse, so checking allowable inputs and outputs prevents invalid substitutions or undefined results.

Key notes

Important points to keep in mind

Function notation f(x) labels the rule and the input; read it as 'f of x'.

Inverse functions reverse inputs and outputs; an inverse exists only for one-to-one functions.

Find an inverse by writing y = f(x), swapping x and y, then solving for y.

Composite f(g(x)) applies g first, then f; substitute g(x) into f directly.

Order matters in composition; always perform the inner function before the outer.

Domain and range swap roles between a function and its inverse.

Use the horizontal line test to check one-to-one behaviour on a graph.

Restrict a function's domain when necessary to create a valid inverse.

Check inverses by composing both ways and confirming the result simplifies to x.

When composing, ensure the inner function output lies within the outer function's domain.