2.2 Concept of a function, domain, range and inverseIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A function is defined by $f(x)=\sqrt{x-3}$.

   (a)

   Which of the following is the largest possible domain of $f$?

   [1 mark]

   • A$x>3$
   • B$x\ge0$
   • C$x\ge3$
   • D$x\le3$

   (b)

   Which of the following is the range of $f$, for its largest possible domain?

   [1 mark]

   • A$f(x)\ge3$
   • B$f(x)>0$
   • Call real numbers
   • D$f(x)\ge0$

   (c)

   Find the value of $x$ for which $f(x)=4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A taxi company charges a fixed fee of 12 AED plus 2.5 AED for each kilometre travelled. The cost, $C$ AED, of a journey of $n$ km is modelled by $C(n)=12+2.5n$, for $n\ge0$.

   (a)

   Find the cost of an 18 km journey.

   [1 mark]

   • A$57$ AED
   • B$261$ AED
   • C$30$ AED
   • D$45$ AED

   (b)

   What does $C^{-1}(100)$ represent in this context?

   [1 mark]

   • AThe cost of a 100 km journey
   • BThe distance, in km, that can be travelled for 100 AED
   • CThe cost per kilometre of a 100 km journey
   • DThe fixed fee when the total cost is 100 AED

   (c)

   Find the value of $C^{-1}(100)$, giving the units of your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $g$ is defined by $g(x)=2x+5$ for $-1\le x\le4$.

   (a)

   Find the range of $g$.

   [3 marks]

   (b)

   (i) Find $g^{-1}(x)$.
   (ii) Write down the domain of $g^{-1}$.
   (iii) Find $g^{-1}(9)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ball is thrown upwards from a platform. Its height above the ground, $h$ metres, $t$ seconds after it is thrown is modelled by $h(t)=-5t^2+20t+25$ until the ball hits the ground. Use your GDC where appropriate.

   (a)

   (i) Find $h(0)$ and interpret its value.
   (ii) Find the largest possible domain of $h$ in this context.
   (iii) Find the range of $h$.

   [6 marks]

   (b)

   (i) Explain why $h$ has no inverse function on the domain $0\le t\le5$.
   The domain is now restricted to $2\le t\le5$ so that an inverse function $h^{-1}$ exists.
   (ii) Write down the domain and the range of $h^{-1}$.
   (iii) Find $h^{-1}(30)$ and interpret your answer in context.

   [6 marks]

   Total for question 4: 12 marks

End of questions