Functions: domain, range and compositionEdexcel International A Level Maths: Subtopic test

1. 1
   A function $f$ is defined by $f(x)=x^2-4x+7$, $x\in\mathbb{R}$.

   (a)

   Which statement about $f$ is correct?

   [1 mark]

   • AIt is one-one, because every value of $x$ has exactly one image.
   • BIt is one-one, because $f$ is defined for every real $x$.
   • CIt is many-one, because $f(1)=f(3)$.
   • DIt is neither one-one nor many-one, because $f$ is not linear.

   (b)

   Which is the range of $f$?

   [1 mark]

   • A$f(x)\geqslant7$
   • B$f(x)\geqslant-3$
   • C$f(x)\in\mathbb{R}$
   • D$f(x)\geqslant3$

   (c)

   The domain of $f$ is now restricted to $x\geqslant4$. Find the range of $f$ on this domain.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Which expression is $\mathrm{fg}(x)$?

   [1 mark]

   • A$2x^2+1$
   • B$4x^2+12x+8$
   • C$2x^2+3$
   • D$2x^2-1$

   (b)

   Find the value of $\mathrm{gf}(1)$.

   [1 mark]

   • A$3$
   • B$24$
   • C$5$
   • D$26$

   (c)

   Solve $\mathrm{fg}(x)=19$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The functions $f$ and $g$ are defined by $f(x)=\sqrt{x-2}+1$, $x\geqslant2$, and $g(x)=\dfrac{4}{x-1}$, $x\in\mathbb{R}$, $x\neq1$.

   (a)

   State the range of $f$ and find an expression for $\mathrm{gf}(x)$, stating its domain.

   [3 marks]

   (b)

   Find the largest possible domain of $\mathrm{fg}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f$ and $g$ are defined by $f(x)=3-\dfrac{2}{x}$, $x\in\mathbb{R}$, $x\neq0$, and $g(x)=x^2+1$, $x\in\mathbb{R}$.

   (a)

   (i) State the range of $f$, explaining why $f$ cannot take one value.
   (ii) Explain why $f$ is a one-one function.
   (iii) Find $\mathrm{fg}(x)$.
   (iv) Find the range of $\mathrm{fg}$.

   [6 marks]

   (b)

   (i) Find $\mathrm{gf}(x)$.
   (ii) Solve $\mathrm{gf}(x)=5$.

   [6 marks]

   Total for question 4: 12 marks

End of questions