Inverse functions and their graphsEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=\dfrac{2x+1}{x-3}$, $x\in\mathbb{R}$, $x\neq3$.

   (a)

   Which expression is $f^{-1}(x)$?

   [1 mark]

   • A$\frac{x-3}{2x+1}$
   • B$\frac{3x-1}{x+2}$
   • C$\frac{3x+1}{x+2}$
   • D$\frac{3x+1}{x-2}$

   (b)

   Which is the domain of $f^{-1}$?

   [1 mark]

   • A$x\in\mathbb{R}$, $x\neq2$
   • B$x\in\mathbb{R}$, $x\neq3$
   • C$x\in\mathbb{R}$
   • D$x\in\mathbb{R}$, $x\neq-\frac12$

   (c)

   Find the value of $f^{-1}(5)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x)=x^2-6x+5$, $x\geqslant3$.

   (a)

   Which expression is $g^{-1}(x)$?

   [1 mark]

   • A$3-\sqrt{x+4}$
   • B$\sqrt{x+4}-3$
   • C$3+\sqrt{x+4}$
   • D$3+\sqrt{x-4}$

   (b)

   Which is the range of $g^{-1}$?

   [1 mark]

   • A$g^{-1}(x)\geqslant-4$
   • B$g^{-1}(x)\geqslant3$
   • C$g^{-1}(x)\geqslant5$
   • D$g^{-1}(x)\in\mathbb{R}$

   (c)

   Explain why $g$ has an inverse function with the domain $x\geqslant3$, but would not have one if the domain were $x\in\mathbb{R}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=2\mathrm{e}^{x}-5$, $x\in\mathbb{R}$.

   (a)

   Find $f^{-1}(x)$ and state its domain.

   [3 marks]

   (b)

   The graphs of $y=f(x)$ and $y=f^{-1}(x)$ are drawn on the same axes. State the geometrical relationship between them, and find the coordinates of the point where each graph crosses the $y$-axis.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f$ and $g$ are defined by $f(x)=\dfrac{x+3}{x-1}$, $x\in\mathbb{R}$, $x\neq1$, and $g(x)=x^2-4x$, $x\geqslant2$.

   (a)

   (i) Show that $f^{-1}(x)=f(x)$.
   (ii) Write down $\mathrm{ff}(x)$.
   (iii) Find the coordinates of the points where the graph of $y=f(x)$ meets the line $y=x$.

   [6 marks]

   (b)

   (i) State the range of $g$.
   (ii) Find $g^{-1}(x)$ and state its domain.
   (iii) Solve $g(x)=g^{-1}(x)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions