1. ## Composite Function

hi!

Attached is a function qns......now studying for a test and come across problems which i have difficulty completing.... thanks for ur help in advance!

2. A function $f$ is defined by:

$f:x\mapsto ax+b,\ x \in \mathbb{R}$,

where $a$ and $b$ are positive constants.

(a) i. Find $f^3(x),$ (where $f^3(x)=fff(x)$).

$f^2(x) = f(f(f(x))) = f(f(ax+b))$

$=f(a(ax+b)+b)$
$=a(a(ax+b)+b)+b$
$=a^3x+a^2b+ab+b$
(ii) Given that $f^3(x)=64x+21$, find the values of $a$ and $b$.

From part (i) we know that the coefficient of $x$ is $a^3$, so we have $a^3=64$. or $a=4$.

We also know that the constant term is $a^2b+ab+b=21$, which when we substitute in the known value of $a$ gives us $b=1$.

RonL

(other parts to follow in another post if someone else does not provide their solutions first).

3. Originally Posted by CaptainBlack
A function $f$ is defined by:

$f:x\mapsto ax+b,\ x \in \mathbb{R}$,

where $a$ and $b$ are positive constants.
(iii) With the values of $a$ and $b$ found in part (ii) determin $f^n(x)$, leaving your answer in the form $p^nx+\frac{p^n-1}{q}$ where $p$ and $q$ are integers to be found.

From the way that part (i) went I will assume that:

$
f^n(x)=a^nx + a^{n-1}b + \dots +ab + b
$

Putting in $a=4$ and $b=1$:

$f^n(x)=4^n x + 4^{n+1}+\cdots + 4 + 1$

Which simplifies to:

$
f^n(x)=4^nx + \frac{4^n-1}{3}$

RonL

4. Another function $g$ is defined by $g: x\mapsto e^x,\ x \in \mathbb{R},\ x<0$.

(b) (i) Find in terms of $a$ and $b$ the range of $fg$.

With the definition of $g$ the range of $g$ is the open interval $(0,1)$, which $f$ will map to the open interval $(b, (b+a))$

(ii) If $h(x)=[g(x)]^2$, determine, with a reason, whether $h^{-1}$ exists.

With the give devinition of $g(x)$ we have $h(x)=e^{2x}$, and the range of $h$ is the open interval $(0,1)$. But $\ln$ is defined everywhere on this interval and single valued, so if:

$h(x)=y$,

then $x=\ln(y)/2$, and is the only such $x$ with image under $h$ of $y$

RonL

5. thanks for ur help !