Composite and inverse functionsEdexcel A-Level Maths: Subtopic test

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

   (a)

   Find the value of $fg(2)$.

   [1 mark]

   • A$17$
   • B$13$
   • C$20$
   • D$9$

   (b)

   Find an expression for $gf(x)$.

   [1 mark]

   • A$3x^2+1$
   • B$9x^2+1$
   • C$9x^2-12x+4$
   • D$9x^2-12x+5$

   (c)

   Solve the equation $fg(x)=49$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find an expression for $h^{-1}(x)$.

   [1 mark]

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

   (b)

   Find the range of $h$.

   [1 mark]

   • A$h(x)\in\mathbb{R}$, $h(x)\neq3$
   • B$h(x)>2$
   • C$h(x)\in\mathbb{R}$, $h(x)\neq2$
   • D$h(x)\in\mathbb{R}$, $h(x)\neq-\frac12$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=x^2-4x+7$, $x\geq2$.

   (a)

   Express $f(x)$ in the form $(x-a)^2+b$ and hence state the range of $f$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f$ and $g$ are defined by $f(x)=e^{2x}-3$, $x\in\mathbb{R}$, and $g(x)=\ln(x+4)$, $x>-4$.

   (a)

   (i) State the range of $f$.
   (ii) Find $f^{-1}(x)$.
   (iii) Find $fg(x)$, giving your answer as a quadratic in $x$.

   [6 marks]

   (b)

   (i) Solve $gf(x)=\ln5$, giving your answer in exact form.
   (ii) Explain why $gf$ is defined for all real $x$, and state the range of $gf$.

   [6 marks]

   Total for question 4: 12 marks

End of questions