Modulus of functions and reciprocal graphsAQA A-Level Further Maths: Subtopic test

1. 1
   Let $\mathrm{f}(x)=|2x-1|$.

   (a)

   Solve $\mathrm{f}(x)=5$.

   [1 mark]

   • A$x=3$ or $x=-2$
   • B$x=3$ or $x=-3$
   • C$x=2$ or $x=-3$
   • D$x=3$ only

   (b)

   Solve $\mathrm{f}(x)<3$.

   [1 mark]

   • A$x<2$
   • B$x<-1$ or $x>2$
   • C$-2<x<2$
   • D$-1<x<2$

   (c)

   Solve $\mathrm{f}(x)=x+4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\mathrm{f}(x)=x^2+2x-3$ and $\mathrm{g}(x)=\dfrac{1}{\mathrm{f}(x)}$.

   (a)

   Which are the vertical asymptotes of $y=\mathrm{g}(x)$?

   [1 mark]

   • A$x=1$ and $x=-3$
   • B$x=-1$ and $x=3$
   • C$x=-1$ only
   • D$y=0$ only

   (b)

   What is the $y$-intercept of $y=\mathrm{g}(x)$?

   [1 mark]

   • A$-3$
   • B$\frac13$
   • C$-\frac13$
   • D$3$

   (c)

   Find the coordinates of the turning point of $y=\mathrm{g}(x)$ and state its nature.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\mathrm{f}(x)=x^2-4x$.

   (a)

   Solve $|\mathrm{f}(x)|=4$.

   [3 marks]

   (b)

   Solve $|\mathrm{f}(x)|<5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{f}(x)=x^2-6x+5=(x-1)(x-5)$.

   (a)

   (i) Solve $|\mathrm{f}(x)|=3$.
   (ii) Find the values of $k$ for which $|\mathrm{f}(x)|=k$ has exactly four solutions.

   [6 marks]

   (b)

   Consider the curve $y=\dfrac{1}{\mathrm{f}(x)}$.
   (i) Write down the equations of all the asymptotes.
   (ii) Find the coordinates of the turning point and state its nature.
   (iii) Hence write down the range of $\dfrac{1}{\mathrm{f}(x)}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions