Edexcel PMT Mocks — Question 10 6 marks

Exam BoardEdexcel
ModulePMT Mocks (PMT Mocks)
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeEvaluate composite at point
DifficultyModerate -0.8 This is a straightforward multi-part question on composite and inverse functions requiring only routine techniques: reading range from a quadratic, evaluating a composition at a point (substitute g(3) into f), and finding an inverse of a simple rational function. All parts are standard textbook exercises with no problem-solving or novel insight required.
Spec1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence
10. The functions f and g are defined with their respective domains by $$\begin{array} { l l l } \mathrm { f } ( x ) = 4 - x ^ { 2 } & x \in R & x \geq 0 \\ \mathrm {~g} ( x ) = \frac { 2 } { x + 1 } & x \in R & x \geq 0 \end{array}$$ a. Write down the range of f .
b. Find the value of \(\mathrm { fg } ( 3 )\) c. Find \(\mathrm { g } ^ { - 1 } ( x )\)
(6 marks)
AnswerMarks
Part a.(1 mark)
B1Correct range. Look for \(f(x) \leq 4\). Allow equivalent notation e.g. \(y \leq 4\), \(f \leq 4\), \(y \in (-\infty, 4]\)
AnswerMarks
Part b.(2 marks)
M1Full attempt at method to find fg(3) condoning slips. Implied by a correct answer or 3.75
Example: For a correct order of operations so require an attempt to apply g(3) first and then f to their g(3)
Example: \(g(3) = \frac{2}{3+1} = 0.5\) \(\Rightarrow f(0.5) = 4 - 0.5^2 = 3.75\)
AnswerMarks
A1Correct exact value. 3.75 or \(3\frac{3}{4}\) or \(\frac{15}{4}\)
AnswerMarks
Part c.(3 marks)
M1Changes the subject of \(y = \frac{2}{x+1}\) and obtains \(x = \frac{2y}{y}\) or \(x = \frac{2}{y} \pm 1\) or equivalent
Alternatively changes the subject of \(x = \frac{2}{y+1}\) and obtains \(y = \frac{2x}{x}\) or \(y = \frac{2}{x} \pm 1\) or equivalent.
AnswerMarks
A1\(g^{-1}(x) = \frac{2-x}{x}\); \(g^{-1}(x) = \frac{2}{x} - 1\)
Condone \(y = \frac{2}{x} - 1\)
AnswerMarks Guidance
B1Correct domain. \(0 < x \leq 2\) Allow equivalent notation e.g. \(x \in (0, 2]\) 
(6 marks)

**Part a.** | (1 mark)

**B1** | Correct range. Look for $f(x) \leq 4$. Allow equivalent notation e.g. $y \leq 4$, $f \leq 4$, $y \in (-\infty, 4]$

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**Part b.** | (2 marks)

**M1** | Full attempt at method to find fg(3) condoning slips. Implied by a correct answer or 3.75

Example: For a correct order of operations so require an attempt to apply g(3) first and then f to their g(3)

Example: $g(3) = \frac{2}{3+1} = 0.5$ $\Rightarrow f(0.5) = 4 - 0.5^2 = 3.75$

**A1** | Correct exact value. 3.75 or $3\frac{3}{4}$ or $\frac{15}{4}$

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**Part c.** | (3 marks)

**M1** | Changes the subject of $y = \frac{2}{x+1}$ and obtains $x = \frac{2y}{y}$ or $x = \frac{2}{y} \pm 1$ or equivalent

Alternatively changes the subject of $x = \frac{2}{y+1}$ and obtains $y = \frac{2x}{x}$ or $y = \frac{2}{x} \pm 1$ or equivalent.

**A1** | $g^{-1}(x) = \frac{2-x}{x}$; $g^{-1}(x) = \frac{2}{x} - 1$

Condone $y = \frac{2}{x} - 1$

**B1** | Correct domain. $0 < x \leq 2$ | Allow equivalent notation e.g. $x \in (0, 2]$

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10. The functions f and g are defined with their respective domains by

$$\begin{array} { l l l } 
\mathrm { f } ( x ) = 4 - x ^ { 2 } & x \in R & x \geq 0 \\
\mathrm {~g} ( x ) = \frac { 2 } { x + 1 } & x \in R & x \geq 0
\end{array}$$

a. Write down the range of f .\\
b. Find the value of $\mathrm { fg } ( 3 )$\\
c. Find $\mathrm { g } ^ { - 1 } ( x )$\\

\hfill \mbox{\textit{Edexcel PMT Mocks  Q10 [6]}}
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