Composite and inverse functionsAQA A-Level Maths: Subtopic test

1. 1
   The functions $f$ and $g$ are defined for all real $x$ by $f(x)=3x-2$ and $g(x)=x^2+1$.

   (a)

   Find $fg(x)$.

   [1 mark]

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

   (b)

   Find the value of $gf(2)$.

   [1 mark]

   • A$13$
   • B$20$
   • C$5$
   • D$17$

   (c)

   Solve $fg(x)=13$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

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

   [1 mark]

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

   (b)

   State the largest possible domain of $h^{-1}$.

   [1 mark]

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

   (c)

   Given that $h^{-1}(a)=5$, find the value of $a$ without finding $h^{-1}(x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=(x-2)^2+3$ for $x\ge2$.

   (a)

   State the range of $f$ and find $f^{-1}(x)$.

   [3 marks]

   (b)

   State the domain of $f^{-1}$. By considering the relationship between the graphs of $y=f(x)$ and $y=f^{-1}(x)$, show that these graphs do not meet.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f$ and $g$ are defined by $f(x)=\sqrt{x+5}$ for $x\ge-5$ and $g(x)=x^2-4$ for $x\in\mathbb{R}$.

   (a)

   Find $gf(x)$, stating its domain and range, and find $fg(x)$, stating its range.

   [6 marks]

   (b)

   Find $f^{-1}(x)$, stating its domain and range, and find the exact value of $x$ for which $f(x)=f^{-1}(x)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions