A function’s domain is the set of allowed inputs, a composition f(g(x)) feeds the output of g into f, and an inverse function undoes f by swapping inputs and outputs. Only one-to-one functions have inverses, so a function such as x squared needs a restricted domain like x at least 0 first.
Key facts:
- Standards F-IF.A.1, F-BF.A.1c, F-BF.B.4b, F-BF.B.4c and F-BF.B.4d: notation, composition, inverses and restricted domains.
- Domain rules: no zero denominators, no negative numbers under even roots, and only positive inputs for \(\ln\).
- In \(f(g(x))\), the input must be in the domain of \(g\) and \(g(x)\) must be in the domain of \(f\).
- \(f^{-1}(f(x))=x\) and \(f(f^{-1}(x))=x\); the domain of \(f\) is the range of \(f^{-1}\) and the reverse.
- A function has an inverse function only if it passes the horizontal line test; restrict the domain when it fails.
What are function notation, domain and range?
The symbol \(f(x)\) names the output of function \(f\) for the input \(x\). The domain is every input that gives a real output, and the range is every output the function produces. Two rules shrink a domain: never divide by zero, and never take an even root (or a logarithm) of a quantity that is not allowed.
A square bracket includes an endpoint and a parenthesis excludes it. \([3,\infty)\) means \(x \ge 3\); \((-\infty,2)\) means \(x < 2\). The symbol ∪ joins separate pieces, so \(x \neq 4\) becomes \((-\infty,4)\) ∪ \((4,\infty)\).
How do you compose functions?
The composition \(f(g(x))\) (also written f ∘ g) feeds the output of \(g\) into \(f\). Its domain contains each \(x\) that is allowed in \(g\) and whose value \(g(x)\) is allowed in \(f\).
With \(f(x)=x^{2}+1\) and \(g(x)=2x-1\), we get \(f(g(x))=(2x-1)^{2}+1=4x^{2}-4x+2\) but \(g(f(x))=2x^{2}+1\). To decompose a function, find an inner piece and an outer piece, for example \(h(x)=\sqrt{x^{2}+9}\) is \(f(x)=\sqrt{x}\) after \(g(x)=x^{2}+9\).
How do you find an inverse function?
An inverse function \(f^{-1}\) undoes \(f\): if \(f(a)=b\) then \(f^{-1}(b)=a\). Its graph is the mirror image of the graph of \(f\) across \(y=x\).
- Write \(y=f(x)\).
- Swap \(x\) and \(y\).
- Solve the new equation for \(y\).
- Name the result \(f^{-1}(x)\) and verify that \(f^{-1}(f(x))=x\).
Example: for \(f(x)=\frac{x}{x+2}\) swap to get \(x=\frac{y}{y+2}\), so \(xy+2x=y\) and \(y=\frac{2x}{1-x}\). Check: \(f(1)=\frac{1}{3}\) and \(f^{-1}(\frac{1}{3})=1\).
Why must a domain sometimes be restricted?
Only one-to-one functions have inverse functions. The horizontal line test checks this: a horizontal line must never meet the graph twice.
The parabola \(y=x^{2}\) fails the test, but keeping only \(x \ge 0\) passes it and gives the inverse \(y=\sqrt{x}\).
Functions: Domain, Range, Composition and Inverses: frequently asked questions
How do you write a domain in interval notation?
List the allowed x-values from left to right, using brackets for included endpoints and parentheses for excluded ones or for infinity. For \(f(x)=\sqrt{x-1}\) the domain is \([1,\infty)\).
Is f(g(x)) the same as g(f(x))?
Usually not. With \(f(x)=x+3\) and \(g(x)=2x\), \(f(g(x))=2x+3\) but \(g(f(x))=2x+6\).
How do you check that two functions are inverses?
Show that \(f(g(x))=x\) and \(g(f(x))=x\) for every x in the proper domains. One composition alone is not enough to be sure.
Why does y = x squared have no inverse function?
Two different inputs such as \(-2\) and \(2\) give the same output \(4\), so reversing the rule would give two answers. Restricting to \(x \ge 0\) fixes that.
How do you read an inverse value from a graph or table?
To find \(f^{-1}(b)\), locate the output \(b\) and read the matching input. If the table shows \(f(4)=9\), then \(f^{-1}(9)=4\).
See every chapter on the Precalculus math page.