What is a one-one function?

One where every output comes from exactly one input.

What is a many-one function?

One where at least two inputs give the same output.

Is a one-to-many rule a function?

No. A function gives one output per input.

When the range of g lies within the domain of f.

How can a domain be restricted to make a function one-one?

Choose a part where the function is always increasing or always decreasing.

Functions: domain, range and compositionEdexcel International A Level Maths: Flashcards

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What is a function?: A mapping that gives exactly one output for each input in its domain.
What is the domain?: The set of allowed inputs.
What is the range?: The set of outputs the function produces.
What is a one-one function?: One where every output comes from exactly one input.
What is a many-one function?: One where at least two inputs give the same output.
Is a one-to-many rule a function?: No. A function gives one output per input.
Range of $f(x)=(x-2)^2+3$ , $x\in\mathbb{R}$ ?: $f(x)\geqslant3$
What does $\mathrm{fg}(x)$ mean?: $f(g(x))$ : do $g$ first, then $f$ .
$f(x)=2x+3$ , $g(x)=x^2-1$ : find $\mathrm{fg}(x)$ .: $2x^2+1$
$f(x)=2x+3$ , $g(x)=x^2-1$ : find $\mathrm{gf}(x)$ .: $(2x+3)^2-1=4x^2+12x+8$
Is $\mathrm{fg}=\mathrm{gf}$ in general?: No, the order matters.
When is $\mathrm{fg}$ defined?: When the range of $g$ lies within the domain of $f$ .
How can a domain be restricted to make a function one-one?: Choose a part where the function is always increasing or always decreasing.

Exam questions on Functions: domain, range and composition

A function $f$ is defined by $f(x)=x^2-4x+7$ , $x\in\mathbb{R}$ .
The domain of $f$ is now restricted to $x\geqslant4$ . Find the range of $f$ on this domain.2 marks
The functions $f$ and $g$ are defined by $f(x)=2x+3$ , $x\in\mathbb{R}$ , and $g(x)=x^2-1$ , $x\in\mathbb{R}$ .
Solve $\mathrm{fg}(x)=19$ .2 marks
The functions $f$ and $g$ are defined by $f(x)=\sqrt{x-2}+1$ , $x\geqslant2$ , and $g(x)=\dfrac{4}{x-1}$ , $x\in\mathbb{R}$ , $x\neq1$ .
State the range of $f$ and find an expression for $\mathrm{gf}(x)$ , stating its domain.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).