The modulus functionEdexcel A-Level Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=|2x-1|$.

   (a)

   What is the value of $f(-2)$?

   [1 mark]

   • A$-5$
   • B$5$
   • C$3$
   • D$-3$

   (b)

   What are the coordinates of the vertex of the graph of $y=|2x-1|$?

   [1 mark]

   • A$\left(-\frac12,0\right)$
   • B$(0,1)$
   • C$\left(\frac12,0\right)$
   • D$\left(\frac12,1\right)$

   (c)

   Solve $f(x)=3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x)=|3x+2|$.

   (a)

   Which set of values of $x$ satisfies $g(x)<4$?

   [1 mark]

   • A$x<-2$ or $x>\frac23$
   • B$-\frac23<x<2$
   • C$-\frac43<x<\frac43$
   • D$-2<x<\frac23$

   (b)

   Which set of values of $x$ satisfies $g(x)>4$?

   [1 mark]

   • A$x<-2$ or $x>\frac23$
   • B$-2<x<\frac23$
   • C$x<-2$ and $x>\frac23$
   • D$x>\frac23$

   (c)

   State the coordinates where the graph of $y=g(x)$ meets the $y$-axis, and the coordinates of its vertex.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the graph of $y=|2x-1|$ and the straight line $y=x$.

   (a)

   Solve $|2x-1|=x$.

   [3 marks]

   (b)

   Hence solve $|2x-1|>x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=|3x-6|$, and let $L_k$ be the line $y=x+k$, where $k$ is a constant.

   (a)

   In this part take $k=2$, so the line is $y=x+2$.
   (i) State the coordinates of the points where the graph of $y=f(x)$ meets the axes.
   (ii) Solve $|3x-6|=x+2$.
   (iii) Hence solve $|3x-6|<x+2$.

   [6 marks]

   (b)

   (i) Show that the solutions of $|3x-6|=x+k$, where they exist, are $x=\frac{k+6}{2}$ and $x=\frac{6-k}{4}$.
   (ii) The vertex of the graph of $y=f(x)$ is at $(2,0)$. Use this to find the set of values of $k$ for which the equation $|3x-6|=x+k$ has two solutions.
   (iii) Find the value of $k$ for which the equation has exactly one solution, and state that solution.

   [6 marks]

   Total for question 4: 12 marks

End of questions