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

1. 1
   Consider the functions $f(x)=\operatorname{sech}x$ and $g(x)=\operatorname{cosech}x$, each defined on its largest possible domain.

   (a)

   What is the range of $f$?

   [1 mark]

   • A$0\le y\le1$
   • B$-1<y<1$
   • C$0<y\le1$
   • D$y\ge1$

   (b)

   What is the largest possible domain of $g$?

   [1 mark]

   • Aall real $x$ except $x=0$
   • Ball real $x$
   • C$x>0$
   • Dall real $x$ except $x=1$

   (c)

   State the range of $g$, and explain why $g$ is not defined at $x=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $h$ is defined by $h(x)=\coth x$ on its largest possible domain.

   (a)

   What is the range of $h$?

   [1 mark]

   • A$-1<y<1$
   • B$y<-1$ or $y>1$
   • C$y>1$
   • Dall real $y$ except $y=0$

   (b)

   Which statement describes $\coth x$ as $x\to\infty$?

   [1 mark]

   • Ait approaches $0$
   • Bit increases without limit
   • Cit approaches $1$ from below
   • Dit approaches $1$ from above

   (c)

   Find the exact value of $h(\ln3)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=\cosh x$ for $x\ge0$.

   (a)

   State the range of $f$. Explain why $f$ has an inverse function, and state the domain and range of $f^{-1}$.

   [3 marks]

   (b)

   Find the exact value of $f^{-1}(\sqrt2)$. Explain why $f^{-1}\left(\frac12\right)$ does not exist.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f(x)=\tanh x$ and $g(x)=\operatorname{sech}x$ are defined for all real $x$.

   (a)

   (i) Show that $\operatorname{sech}x=\frac{2}{e^x+e^{-x}}$.
   (ii) Find the range of $g$, giving a reason.
   (iii) Solve $\operatorname{sech}x=\frac12$, giving exact answers.
   (iv) Explain why $\operatorname{sech}x=2$ has no real solution.

   [6 marks]

   (b)

   (i) Explain why $f$ has an inverse function, and state the domain and range of $f^{-1}=\tanh^{-1}$.
   (ii) Find the exact value of $\tanh^{-1}\left(\frac45\right)$ as a single natural logarithm.

   [6 marks]

   Total for question 4: 12 marks

End of questions