Domains, ranges and reciprocal hyperbolic functionsAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

Range of \sinh x?
Range of \cosh x?
Range of \tanh x?
Definition of \operatorname{sech}x and its range?
Definition of \operatorname{cosech}x, its domain and range?
Definition of \coth x?
Domain and range of \coth x?
Why is \operatorname{cosech}x undefined at x=0?
Domain and range of \sinh^{-1}x?
Domain and range of \cosh^{-1}x?
Domain and range of \tanh^{-1}x?
Why must \cosh x be restricted to x\ge0 to have an inverse?

Exam questions on Domains, ranges and reciprocal hyperbolic functions

Consider the functions $f(x)=\operatorname{sech}x$ and $g(x)=\operatorname{cosech}x$ , each defined on its largest possible domain.
State the range of $g$ , and explain why $g$ is not defined at $x=0$ .2 marks
The function $h$ is defined by $h(x)=\coth x$ on its largest possible domain.
Find the exact value of $h(\ln3)$ .2 marks
The function $f$ is defined by $f(x)=\cosh x$ for $x\ge0$ .
State the range of $f$ . Explain why $f$ has an inverse function, and state the domain and range of $f^{-1}$ .3 marks

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).