Domain and rangeIB MYP Maths Extended: Subtopic test

1. 1
   A function is defined by $f(x)=2x-3$, where $x$ is a real number and $-1\leq x\leq4$.

   (a)

   Find the range of $f$.

   [1 mark]

   • A$-5\leq f(x)\leq5$
   • B$-1\leq f(x)\leq4$
   • C$-5\leq f(x)\leq4$
   • D$-3\leq f(x)\leq5$

   (b)

   Which is the domain of $f$ in set notation?

   [1 mark]

   • A$\{x\in\mathbb{R}:-1<x<4\}$
   • B$\{x\in\mathbb{R}:x\geq-1\}$
   • C$\{x\in\mathbb{R}:-1\leq x\leq4\}$
   • D$\{-1,4\}$

   (c)

   The domain of $f$ is restricted so that $f(x)\geq0$. Find the new domain.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the functions $h(x)=\sqrt{x+3}$ and $k(x)=\dfrac{4}{x-5}$. Each function has the largest possible domain of real numbers.

   (a)

   What is the domain of $h$?

   [1 mark]

   • A$x>-3$
   • B$x\geq0$
   • C$x\leq-3$
   • D$x\geq-3$

   (b)

   What is the domain of $k$?

   [1 mark]

   • A$x\neq0$
   • B$x\neq5$
   • C$x>5$
   • D$x\neq-5$

   (c)

   State the range of $h$ and give a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A quadratic function is defined by $f(x)=x^2-6x+5$.

   (a)

   By completing the square, find the coordinates of the lowest point of the graph, and hence state the range of $f$ when its domain is all real numbers.

   [3 marks]

   (b)

   The domain is now restricted to $0\leq x\leq5$. Find the range of $f$ on this domain.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A market stall in Istanbul sells scarves at a price of $p$ dirhams each. The stall owner models the number sold per week as $80-2p$, so the weekly revenue is $R(p)=p(80-2p)$ dirhams.

   (a)

   (i) Explain why the domain of $R$ in this context is $0\leq p\leq40$.
   (ii) Find the range of $R$ on this domain.

   [6 marks]

   (b)

   The owner decides to charge a whole number of dirhams, from $10$ to $35$ inclusive.
   (i) Find the range of $R$ for these prices.
   (ii) The owner says: ‘The model shows $p=20$ is always the best price.’ Evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions