2.7 Composite and inverse functionsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   $f(x)=2x+3$ and $g(x)=x^2$, for $x\in\mathbb{R}$.

   (a)

   Find $(f\circ g)(x)$.

   [1 mark]

   • A$2x^2+3$
   • B$(2x+3)^2$
   • C$4x^2+3$
   • D$x^2+2x+3$

   (b)

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

   [1 mark]

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

   (c)

   Solve $(f\circ g)(x)=11$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The fare in EUR for a taxi journey of $d$ km is $f(d)=3+1.2d$, for $d\geq0$. Fares are converted to AED using $g(x)=4x$.

   (a)

   Which expression gives the fare in AED for a journey of $d$ km?

   [1 mark]

   • A$3+4.8d$
   • B$12+4.8d$
   • C$12+1.2d$
   • D$7+1.2d$

   (b)

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

   [1 mark]

   • A$1.2x+3$
   • B$\frac{x+3}{1.2}$
   • C$\frac{x}{1.2}-3$
   • D$\frac{x-3}{1.2}$

   (c)

   A passenger is charged 96 AED. Find the distance of the journey.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $f(x)=(x-3)^2-2$, defined for $x\in\mathbb{R}$.

   (a)

   (i) Explain why $f$ has no inverse function on the given domain.
   (ii) State the largest domain of the form $x\geq k$ on which $f$ has an inverse, and the range of $f$ on this domain.

   [3 marks]

   (b)

   The domain of $f$ is now restricted to $x\geq3$. Find $f^{-1}(x)$, and state its domain.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of visitors to a website $d$ days after launch is $V(d)=\dfrac{4000}{1+19e^{-0.5d}}$ for $d\geq0$. The advertising revenue in EUR from $v$ visitors is $R(v)=0.35v$.

   (a)

   (i) Find an expression for $(R\circ V)(d)$.
   (ii) Use your GDC to find the revenue after 6 days.
   (iii) Find the range of $R\circ V$.

   [6 marks]

   (b)

   (i) Find $V^{-1}(v)$, the number of days needed to reach $v$ visitors.
   (ii) Hence find after how many days there are 3000 visitors.

   [6 marks]

   Total for question 4: 12 marks

End of questions