Functions
Functions
3.1 Domain & Range of
functions
A) State
the suitable domain for each of the following
1.
$f(x)=\sqrt{x-4}$
2.
$f(x)=\sqrt{6-2x}$
B)
Determine the range of each of the following functions
1.
$f:x\mapsto x+4$, $0<x<5$
2.
$f:x\mapsto x^{2}+7$,$x\in \Re$
3. $f:x\mapsto 2x-3$,$2<x \le 6$
4. $f:x\mapsto x^{2}-6x$,$0\le x\le 6$
5.
$f :x\mapsto 3\sqrt{x}-4$, $x>0$
6. $f:x\mapsto \sqrt{3x-2}$,$2\le x\le 9$
7. $f :x\mapsto (x-3)^{2}+5$, $4\le x\le 6$
8. $f :x\mapsto (x-3)^{2}+5$, $2\le x\le 6$
C) Use
the method of completing the square to find the range of each of these
functions
1.
$f(x)=x^{2}-6x+10$,$x\in \Re$
2.
$f(x)=x^{2}+7x+1$,$x\in \Re$
3.
$f(x)=x^{2}-3x+4$,$x\in \Re$
4.
$f(x)=2x^{2}+8x+1$,$x\in \Re$
5.
$f(x)=3x^{2}+5x-12$,$x\in \Re$
6.
$f(x)=-x^{2}-6x+12$,$x\in \Re$
3.2 One-to-one & Many-to-one
Functions
Determine
which of the following functions are one-to-one and which are many-to-one
1.
$f(x)=x+3$,$x\in \Re$
2. $f(x)=x^{2}+3$,$x\in
\Re$
3.
$f(x)=\frac{1}{x}$,$x\in \Re$,$x\neq 0$
4. $f(x)=(x-4)^{2}$,$2\le x\le 6$
5. $f(x)=x^{2}-4x$,$0<x<4$
6. $f(x)=x^{2}-4x$,$0<x<2$
7. $f(x)=\frac{2}{x-3}$,$-1<x<2$
3.3 Composite function
Given
that $f(x)=2x+1$,$g(x)=x^{2}$ and $h(x)=\frac{1}{x}$ evaluate each of the
following
1.
$f(3)$
2.
$g(2)$
3. $hg(2)$
4.
$fg(-3)$
5.
$gf(1)$
6.
$gh(-2)$
7.
$hf(4)$
8.
$ff(5)$
9.
$fgh(2)$
10.
$hfg(4)$
The
following $f$,$g$ and $h$ are defined by $f(x)=x^{2}$, $g(x)=\frac{3}{x}$ and
$h(x)=2-x$. Find an expression for each of the following composite functions in
term of $x$.
11. $gh$
12. $hg$
13.
$g^{2}$
14.
$h^{2}$
15.
$ghf$
16.
$hgf$
17.
Given that $f(x)=x-3$,$g(x)=10x$ and $h(x)=\frac{1}{x}$, $x\neq 0$
a) Find the expression of $fgh(x)$
b) solve the equation $fgh(x)=x$
18.
Funtions $f$ and $g$ are defined by $f(x)=3x+5$ and $g(x)=\frac{x-5}{3}$
a) Find the expressions of $f^{2}(x)$ and
$g^{2}(x)$
b) Show that $fg(x)=x$ and $ggf(x)=x$
19. The
functions $f$ and $g$ are defined by $f:x\mapsto \frac{x^{2}}{4}$ for {$x\in
\Re$} and $g:x\mapsto \frac{2}{x-1}$ for {$x\in \Re$,$x\neq 1$}. Find
expressions for the following case stating any values of $x$ for which the
composite functions are undefined
(a)
$fg(x)$
(b)
$gf(x)$
(c)
$f^{2}(x)$
(d)
$g^{2}(x)$
3.4 Inverse Functions
A) Find
the inverse of each of the following functions
1.
$f:x\mapsto 3x+2$,$x\in \Re$
2.
$f:x\mapsto 5x-1$,$x\in \Re$
3.
$f:x\mapsto 4-3x$,$x\in \Re$
4.
$f:x\mapsto \frac{2}{x}$,$x\neq 0$
B) Find
the inverse of each of the following functions, and state the domain on which
each inverse is defined
1.
$f:x\mapsto x^{2}$,$x>2$
2.
$f:x\mapsto \frac{1}{2+x}$,$x>0$
3.
$f:x\mapsto \sqrt{x-2}$,$x>3$
4.
$f:x\mapsto 3x^{2}-1$,$1<x<4$
5.
$f:x\mapsto \frac{1}{x-2}$,$x>3$
6.
$f:x\mapsto \sqrt{3-x}$,$x\le 2$
7. $f:x\mapsto (x-3)^{2}+5$,$4\le x\le 6$
8. $f:x\mapsto
5-\sqrt{x+3}$,$x\ge -3$
C)
1. Given
that $f:x\mapsto 3x-4$,$x\in \Re$
a) Find an expression for the inverse
function $f^{-1}(x)$
b) Sketch the graphs of $f(x)$ and
$f^{-1}(x)$ on the same set of axes
c) Solve the equation $f(x)=f^{-1}(x)$
2. A
function is defined by $f(x)= x^{2}-6$,$x>0$
a) Find an expression for the inverse
function $f^{-1}(x)$
b) Sketch the graphs of $f(x)$ and
$f^{-1}(x)$ on the same set of axes
c) Solve the equation $f(x)=f^{-1}(x)$
3. The
function $g(x)=\frac{x+1}{x-3}$ is defined for $x\in \Re$,$x\neq 3$. Find its
inverse function, and state the value of $x$ for which $g^{-1}(x)$ is not
defined.
4.
Functions $f$ and $g$ are defined by
$f:x\mapsto
3x+2$,$x\in \Re$;$g:x\mapsto \frac{6}{2x+3}$,$x\in \Re$,$x\neq -1.5$
i) Find the value of $x$ for which
$fg(x)=3$
ii) Sketch, in a single diagram, the
graphs of $y=f(x)$ and $y=f^{-1}(x)$, making clear the relationship between
these two graphs
iii) Express each of $f^{-1}(x)$ and
$g^{-1}(x)$ in terms of $x$, and solve the equation $f^{-1}(x)=g^{-1}(x)$
3.5 Mixed Exercise
1. The
linear function $f:x\mapsto ax+b$ is such that $f(1)=8$ and $f(3)=14$. Find $a$
and $b$
2. The
linear function $f:x\mapsto ax+b$ where $a$ and $b$ are constants. Given that
$f(2)=3$ and $f(-3)=13$, find $a$ and $b$. Hence, calculate the value of $c$
for which $f(c)=0$.
3. The
function $m$ is defined by $m:x\mapsto 2+5x$, for $x\ge 1$. Find the range of
$m$. Find the inverse of $m$, and find its domain and range.
4. Given
that $f(x)=x^{2}+3$, and $g(x)=2x+a$ and $fg(x)=4x^{2}-8x+7$, find the value of
the constant $a$.
5. The
functions $f$ and $g$ are defined as follows:
$$f:x\mapsto
x+1,x\in \Re$$ and $$g:x\mapsto x^{2}-3,x\in \Re$$.
i) Show that $fg(x)+gf(x)=2x^{2}+2x-4$
ii) Hence solve $fg(x)+gf(x)=0$
6. The
functions $g$ is defined by $g(x)=2x^{2}-3,x\ge 0$
i) State the range of $g$ and sketch its
graph
ii) Explain why the inverse function
$g^{-1}$ exist and sketch its graph
iii) Given also that $h$ is defined by
$h(x)=\sqrt{5x+2},x\ge -\frac{2}{5}$, solve the inequality $gh(x)\ge x$
7. The
function $g$ is defined by $g:x\mapsto \frac{ax-4}{x+1},x\in \Re ,x\neq 1$.
Find the positive value of $a$ for which there is only one value of $x$
satisfying $g(x)=x$
3.6 More exercise
1. i)
Express $2x^{2}+8x-10$ in the form $a(x+b)^{2}+c$.
ii)
For the curve $y=2x^{2}+8x-10$, state the least value of $y$ and the
corresponding value of $x$
iii)
Find the set of values of $x$ for which $y\ge 14$
Given that $f:x\mapsto 2x^{2}+8x-10$ for
the domain $x\ge k$
iv)
find the least value of $k$ for which $f$ is one-one.
v) express $f^{-1}$ in terms of $x$ in this
case
2. The
equation of a curve is $y=8x-x^{2}$
i) Express $8x-x^{2}$ in the form
$a-(x+b)^{2}$, stating the numerical values of $a$ and $b$
ii) Hence, or otherwise, find the
coordinates of the stationary point of the curve
iii) Find the set of values of $x$ for
which $y\ge -20$
The function $g$ is defined by $g:x\mapsto
8x-x^{2}$, for $x\ge 4$
iv) State the domain and range of $g^{-1}$
v) Find an expression, in terms of $x$,
for $g^{-1}(x)$
3.
Fundtions $f$ and $g$ are defined by
$$f:x\mapsto
2x-5,x\in \Re$$ $$g:x\mapsto
\frac{4}{2-x},x\in \Re, x\neq 2$$
i) Find the value of $x$ for which
$fg(x)=7$
ii) Express each of $f^{-1}(x)$ and
$g^{-1}(x)$ in terms of $x$
iii) Show that the equation
$f^{-1}(x)=g^{-1}(x)$ has no real roots
iv) Sketch, on a single diagram, the
graphs of $y=f(x)$ and $y=f^{-1}(x)$, making clear the relationship between
these two graphs.
4. The
functions $f$ and $g$ are defined as follows:
$$f:x\mapsto
x^{2}-2x,x\in \Re$$ $$g:x\mapsto
2x+3,x\in \Re$$
i) Find the set of values of $x$ for which
$f(x)>15$
ii) Find the range of $f$ and state
whether $f$ has an inverse
iii) Show that the equation $gf(x)=0$ has
no real solutions
iv) Sketch, in a single diagram, the
graphs of $y=g(x)$ and $y=g^{-1}(x)$, making the relationship between the
graphs
5. The
function $f:x\mapsto 2x-a$, where $a$ is a constant, is defined for all real
$x$
i) In the case where $a=3$, solve the
equation $ff(x)=11$
The function $g:x\mapsto x^{2}-6x$ is
defined for all real $x$
ii) Find the value of $a$ for which the
equation $f(x)=g(x)$ has exactly one real solution
The function $h:x\mapsto x^{2}-6x$ is
defined for the domain $x\ge 3$
iii) Express $x^{2}-6x$ in the form
$(x-p)^{2}-q$, where $p$ and $q$ are constants
iv) Find an expression for $h^{-1}(x)$ and
state the domain of $h^{-1}$
6.
Functions $f$ and $g$ are defined by $f:x\mapsto k-x$ for $x\in \Re$, where $k$
is a constant, $g:x\mapsto \frac{9}{x+2}$ for $x\in \Re,x\neq -2$
i) Find the values of $k$ for which the
equation $f(x)=g(x)$ has two equal roots and solve the equation $f(x)=g(x)$ in
these cases
ii) Solve the equation $fg(x)=5$ when
$k=6$
iii) Express $g^{-1}(x)$ in terms of $x$
7. The
function $f$ is defined by $f:x\mapsto x^{2}-3x$ for $x\in \Re$
i) Find the set of values of $x$ for which
$f(x)>4$
ii) Express $f(x)$ in the form
$(x-a)^{2}-b$, stating the values of $a$ and $b$
iii) Write down the range of $f$
iv) State, with a reason, whether $f$ has an
inverse
The function $g$ is defined by $g:x\mapsto
x-3\sqrt{x}$ for $x\ge 0$
v) Solve the equation $g(x)=10$
8.
Determine the set of values of the constant $k$ for which the line $y=4x+k$
does not intersect the curve $y=x^{2}$
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