AQA Paper 1 2019 June — Question 6 8 marks

Exam BoardAQA
ModulePaper 1 (Paper 1)
Year2019
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeFind inverse function
DifficultyModerate -0.3 This is a standard A-level question on inverse functions covering routine techniques: finding range from a restricted domain, deriving an inverse algebraically, and using the reflection property. The algebra is straightforward (solving a quadratic), and all parts follow predictable patterns from textbook exercises. Slightly easier than average due to the simple quadratic form and clear domain restriction, though the intersection point requires solving f(x)=x which adds minor problem-solving.
Spec1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence
The function f is defined by $$f(x) = \frac{1}{2}(x^2 + 1), \quad x \geq 0$$
  1. Find the range of f. [1 mark]
    1. Find \(f^{-1}(x)\) [3 marks]
    2. State the range of \(f^{-1}(x)\) [1 mark]
  3. State the transformation which maps the graph of \(y = f(x)\) onto the graph of \(y = f^{-1}(x)\) [1 mark]
  4. Find the coordinates of the point of intersection of the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) [2 marks]
Question 6:
AnswerMarks
6(a)Deduces the range of f
1
( )≥
Accept f x ,
2
1
y≥ or [0.5,∞)
2
AnswerMarks Guidance
OEAO2.2a B1
2
AnswerMarks
6(b)(i)Rearranges formula, isolating
squared term with at least one
AnswerMarks Guidance
correct step seen.1.1a M1
y = x2 +1
2
2y = x2 +1
2y−1= x2
x= 2y−1
f−1( x )= 2x−1
x ≥
1
Obtains inverse function in any
AnswerMarks Guidance
correct form.1.1b A1
Obtains correct inverse function
f−1(x)=...
using and states
AnswerMarks Guidance
correct domain2.5 A1
(b)(ii) ---
6
AnswerMarks
(b)(ii)States correct range
f−1(x)≥0
Accept
AnswerMarks Guidance
OE1.1b B1
{y: y≥0}
AnswerMarks Guidance
6(c)Recalls correct transformation 1.2
AnswerMarks
6(d)Forms equation using two of the
three expressions
x = =
2
𝑥𝑥 +1
allow their
2 √2𝑥𝑥−1
AnswerMarks Guidance
PI by correct answer3.1a M1
2
𝑥𝑥 +1
2
(1, 1)
Obtains x=√1 2a𝑥𝑥nd− y1=1
AnswerMarks Guidance
CSO1.1b A1
Total8
QMarking instructions AO 
Question 6:
--- 6(a) ---
6(a) | Deduces the range of f
1
( )≥
Accept f x ,
2
1
y≥ or [0.5,∞)
2
OE | AO2.2a | B1 | {y: y≥ 1}
2
--- 6(b)(i) ---
6(b)(i) | Rearranges formula, isolating
squared term with at least one
correct step seen. | 1.1a | M1 | 1( )
y = x2 +1
2
2y = x2 +1
2y−1= x2
x= 2y−1
f−1( x )= 2x−1
x ≥
1
Obtains inverse function in any
correct form. | 1.1b | A1
Obtains correct inverse function
f−1(x)=...
using and states
correct domain | 2.5 | A1
--- 6
(b)(ii) ---
6
(b)(ii) | States correct range
f−1(x)≥0
Accept
OE | 1.1b | B1 | 2
{y: y≥0}
--- 6(c) ---
6(c) | Recalls correct transformation | 1.2 | B1 | Reflection in y = x
--- 6(d) ---
6(d) | Forms equation using two of the
three expressions
x = =
2
𝑥𝑥 +1
allow their
2 √2𝑥𝑥−1
PI by correct answer | 3.1a | M1 | x =
2
𝑥𝑥 +1
2
(1, 1)
Obtains x=√1 2a𝑥𝑥nd− y1=1
CSO | 1.1b | A1
Total | 8
Q | Marking instructions | AO | Mark | Typical solution
The function f is defined by

$$f(x) = \frac{1}{2}(x^2 + 1), \quad x \geq 0$$

\begin{enumerate}[label=(\alph*)]
\item Find the range of f. [1 mark]

\item 
\begin{enumerate}[label=(\roman*)]
\item Find $f^{-1}(x)$ [3 marks]
\item State the range of $f^{-1}(x)$ [1 mark]
\end{enumerate}

\item State the transformation which maps the graph of $y = f(x)$ onto the graph of $y = f^{-1}(x)$ [1 mark]

\item Find the coordinates of the point of intersection of the graphs of $y = f(x)$ and $y = f^{-1}(x)$ [2 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 1 2019 Q6 [8]}}
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