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###### A-Level H2 Math | 5 Essential Questions

# Functions

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In mathematics, the function is a relation that relates the elements from a set (domain) to the elements in another set (codomain). Each element in the domain is related to exactly one element in the codomain. Some of the frequently used functions are quadratic functions, polynomial functions, inverse functions, modulus functions, etc.Â

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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]

- (i)
- (ii)
- (iii)

- (i)
- (ii)
- (iii)

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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,

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

(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]

- (i)
- (ii)
- (iii)

- (i)
- (ii)
- (iii)

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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]

- (i)
- (ii)
- (iii)

- (i)
- (ii)
- (iii)

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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]

- (i)(a)
- (i)(b)
- (ii)
- (iii)(a)
- (iii)(b)

- (i)(a)
- (i)(b)
- (ii)
- (iii)(a)
- (iii)(b)

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(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]

- (i)
- (ii)
- (iii)
- (iv)

- (i)
- (ii)
- (iii)
- (iv)

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