A-Level H2 Math
5 Essential Questions

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2020 YIJC P1 Q7

The functions $\text{f}$ and $\text{g}$ are defined by

$\begin{matrix}
& \text{f}:x\mapsto {{x}^{2}}+3x+1,x\in \mathbb{R},x\le -\frac{3}{2}, \\
& \text{g}:x\mapsto 3+{{\text{e}}^{-x}},x\in \mathbb{R}. \\
\end{matrix}$

(i)

Show that $\text{f}$ has an inverse.

[1]

(i) Show that $\text{f}$ has an inverse.

[1]

(ii)

Give a reason why the composite function ${{\text{f}}^{-1}}\text{g}$ exists. Find ${{\text{f}}^{-1}}\text{g}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}\text{g}$.

[6]

(ii) Give a reason why the composite function ${{\text{f}}^{-1}}\text{g}$ exists. Find ${{\text{f}}^{-1}}\text{g}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}\text{g}$.

[6]

The function $\text{h}$ is defined as follows.

$\text{h}:x\mapsto \ln \left( x-k+1 \right),x\in \mathbb{R},x\ge k,$ where $k<0$.

(iii)

Find the value of $k$ such that the range of $\text{hg}$ is given by $\left( \ln 5,\infty \right)$.

[2]

(iii) Find the value of $k$ such that the range of $\text{hg}$ is given by $\left( \ln 5,\infty \right)$.

[2]

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2020 DHS Promo Q3

Functions $\text{f}$ and $\text{g}$ are defined by

$\text{f}:x\mapsto {{x}^{2}}+cx+d,x\in \mathbb{R},$ where $c$ and $d$ are constants,

 $\text{g}:x\mapsto \ln x,x\in \mathbb{R},x>0$.

(i)

Given that the composite function $\text{gf}$ exists, find an inequality involving $c$ and $d$.

[2]

(i) Given that the composite function $\text{gf}$ exists, find an inequality involving $c$ and $d$.

[2]

For the rest of the question, take $c=2,d=4$.

(ii)

Write down an expression for $\text{gf}\left( x \right)$ and find exactly the range of $\text{gf}$.

[3]

(ii) Write down an expression for $\text{gf}\left( x \right)$ and find exactly the range of $\text{gf}$.

[3]

(iii)

The graph of a function $\text{h}$ is symmetrical about the line $x=k$ if $\text{h}\left( x+k \right)=\text{h}\left( -x+k \right)$ for all valid values of $x$. Using this definition, show that the graph of $\text{gf}$ is symmetrical about the line $x=-1$.

[2]

(iii) The graph of a function $\text{h}$ is symmetrical about the line $x=k$ if $\text{h}\left( x+k \right)=\text{h}\left( -x+k \right)$ for all valid values of $x$. Using this definition, show that the graph of $\text{gf}$ is symmetrical about the line $x=-1$.

[2]

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2020 TMJC Promo Q7
The function $\text{f}$ is defined by

$\text{f}:x\mapsto -\ln (x-2)$, $x\in \mathbb{R}$, $2<x\le 3$.

(i)

Find ${{\text{f}}^{-1}}(x)$ and state the domain and range of ${{\text{f}}^{-1}}$.

[4]

(i) Find ${{\text{f}}^{-1}}(x)$ and state the domain and range of ${{\text{f}}^{-1}}$.

[4]

(ii)

Sketch on the same diagram the graphs of $y=\text{f}(x)$, $y={{\text{f}}^{-1}}(x)$ and $y={{\text{f}}^{-1}}\text{f}(x)$, giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the $x$- and $y$- axes.

[6]

(ii) Sketch on the same diagram the graphs of $y=\text{f}(x)$, $y={{\text{f}}^{-1}}(x)$ and $y={{\text{f}}^{-1}}\text{f}(x)$, giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the $x$- and $y$- axes.

[6]

(iii)

Explain why the x-coordinate of the point of intersection of the graph of $y=\text{f}(x)$ and $y={{\text{f}}^{-1}}(x)$ satisfies the equation

$x+\ln (x-2)=0$,

and find the value of this $x$-coordinate.

[2]

(iii) Explain why the x-coordinate of the point of intersection of the graph of $y=\text{f}(x)$ and $y={{\text{f}}^{-1}}(x)$ satisfies the equation

$x+\ln (x-2)=0$,

and find the value of this $x$-coordinate.

[2]

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2020 EJC Promo Q10

Functions $\text{f}$ and $\text{g}$ are defined by

$\text{f}:x\mapsto {{x}^{3}}-7{{x}^{2}}-5x+11$, $x\in \mathbb{R}$, $x\ge k$,
$\text{g}:x\mapsto {{\left( x+1 \right)}^{2}}+2$,$x\in \mathbb{R}$.

(i)

Let $k=1$.

(i) Let $k=1$.

(a)

Show that $\text{f}$ does not have an inverse.

[2]

(a) Show that $\text{f}$ does not have an inverse.

[2]

(b)

Determine whether the composite function $\text{fg}$ exists.

[2]

(b) Determine whether the composite function $\text{fg}$ exists.

[2]

(ii)

Find the value of $k$ given that ${{\text{f}}^{-1}}$ exists and that the domain of ${{\text{f}}^{-1}}$ is $x\in \mathbb{R}$, $x\ge -24$.

[2]

(ii) Find the value of $k$ given that ${{\text{f}}^{-1}}$ exists and that the domain of ${{\text{f}}^{-1}}$ is $x\in \mathbb{R}$, $x\ge -24$.

[2]

(iii)

Let $k=6$.

(iii) Let $k=6$.

(a)

Show algebraically that $\text{f}'(x)>0$ for all values of $x$ in the domain of $\text{f}$.

[2]

(a) Show algebraically that $\text{f}'(x)>0$ for all values of $x$ in the domain of $\text{f}$.

[2]

(b)

Solve the equation $\text{g}{{\text{f}}^{-1}}(x)=83$.

[3]

(b) Solve the equation $\text{g}{{\text{f}}^{-1}}(x)=83$.

[3]

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2016 ACJC Promo Q8

(i)

A student defines a function $\text{f}$ and its inverse ${{\text{f}}^{-1}}$ by the following:

$\text{f}:x\mapsto \sin x,$ $x\in \mathbb{R},$ $-\pi \le x\le \pi $ ${{\text{f}}^{-1}}:x\mapsto {{\sin }^{-1}}x,$ $x\in \mathbb{R},$ $-1\le x\le 1$.

With the aid of a sketch, explain what is wrong with the definition of $\text{f}$, and suggest a suitable restriction to its domain such that the definition of ${{\text{f}}^{-1}}$ is correct.

[2]

(i) A student defines a function $\text{f}$ and its inverse ${{\text{f}}^{-1}}$ by the following:

$\text{f}:x\mapsto \sin x,$ $x\in \mathbb{R},$ $-\pi \le x\le \pi $
${{\text{f}}^{-1}}:x\mapsto {{\sin }^{-1}}x,$ $x\in \mathbb{R},$ $-1\le x\le 1$.

With the aid of a sketch, explain what is wrong with the definition of $\text{f}$, and suggest a suitable restriction to its domain such that the definition of ${{\text{f}}^{-1}}$ is correct.

[2]

(ii)

The function $\text{g}$ is defined by $\text{g}:x\mapsto {{e}^{-{{\left( x+1 \right)}^{2}}}}$, $x\in \mathbb{R}$, $-1\le x\le 0$. Find the inverse function ${{\text{g}}^{-1}}$ in similar form.

[3]

(ii) The function $\text{g}$ is defined by $\text{g}:x\mapsto {{e}^{-{{\left( x+1 \right)}^{2}}}}$, $x\in \mathbb{R}$, $-1\le x\le 0$. Find the inverse function ${{\text{g}}^{-1}}$ in similar form.

[3]

(iii)

Explain if the composite function ${{\text{g}}^{-1}}{{\text{f}}^{-1}}$ exists.

[1]

(iii) Explain if the composite function ${{\text{g}}^{-1}}{{\text{f}}^{-1}}$ exists.

[1]

(iv)

Write down ${{\text{f}}^{-1}}{{\text{g}}^{-1}}\left( x \right)$and its domain, and find its range. Hence, or otherwise, find the value of $x$ such that $\text{gf}\left( x \right)=\frac{1}{e}$.

[4]

(iv) Write down ${{\text{f}}^{-1}}{{\text{g}}^{-1}}\left( x \right)$and its domain, and find its range. Hence, or otherwise, find the value of $x$ such that $\text{gf}\left( x \right)=\frac{1}{e}$.

[4]

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