Pure Mathematics Practice: Functions

Original practice material

These four structured questions total 42 marks. They practise exam-style pure-mathematics function reasoning suitable for Paper 1 or Paper 2 Section A; they are not a complete paper.

Answers: Functions Answers.

Question 1: Domain, Range and Inverse of a Fractional Function

The function $f$ is defined by

$$f(x)=\frac{2x+1}{x-3},x>3.$$

(a) Show that

$$f(x)=2+\frac{7}{x-3}.$$

Hence state the range of $f$. [3]

(b) Explain why $f$ is one-one on its stated domain. [2]

(c) Find $f^{-1}(x)$ and state its domain. [4]

(d) Hence solve $f(x)=9$. [1]

[Total: 10]

Question 2: Composite Functions and Existence Conditions

Let

$$f:[-2,7]\to \mathbb{R},f(x)=\sqrt{x+2},$$

and

$$g:[0,3]\to \mathbb{R},g(x)=4-x^2.$$

(a) State the range of $f$. [1]

(b) Explain why $gf$ is defined for every $x\in [-2,7]$, and find $gf(x)$. [3]

(c) Find the largest possible domain of $fg$ using the given domain of $g$. [4]

(d) Hence find $fg(x)$ on this domain and state its range. [3]

[Total: 11]

Question 3: Reading a Restricted Graph

The graph below shows

$$h(x)=(x-1{)}^2-2,-1\le x\le 4.$$

(a) State the range of $h$. [2]

(b) Explain why $h$ does not have an inverse function on $[-1,4]$. [2]

The function $k$ is obtained by restricting $h$ to the domain $1\le x\le 4$.

(c) Find $k^{-1}(x)$ and state its domain. [4]

(d) State the range of $k^{-1}$. [1]

[Total: 9]

Question 4: Parameter and One-One Restriction

For a real constant $a$, define

$$p_a(x)=x^2-2ax+3,x\ge a.$$

(a) Write $p_a(x)$ in completed-square form. [2]

(b) Show that $p_a$ is one-one on $x\ge a$. [2]

(c) Find the range of $p_a$ in terms of $a$. [2]

(d) Find $p_a^{-1}(x)$, stating its domain. [4]

(e) Determine the values of $a$ for which the range of $p_a$ is $[-1,\infty )$. [2]

[Total: 12]