Question

33
Simplify the function f(x) =
(81) 4. Then determine the key aspects of the function.
The initial value is
The simplified base is
The domain is
The range is

Answer 1

Answer:

Please check the answer below in detail.

Step-by-step explanation:

The correct form of the function is

$f(x)=\frac{1}{3}(81)^{\frac{3x}{4}$

Lets simplify

$f(x)=\frac{1}{3}\cdot \:81^{\frac{3x}{4}}$

$f(x)=\frac{1}{3}\cdot \:(3^{4}) ^{\frac{3x}{4}}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(3^4\right)^{\frac{3x}{4}}=3^{4\cdot \frac{3x}{4}}=3^{3x}$

So, the function f(x) becomes

$f(x)=\frac{1}{3}\cdot \:3^{3x}$

Use the exponent rule $\frac{a^m}{a^n}=a^{m-n}$

The simplified form of the given function would be:

$f(x)=3^{3x-1}$

As domain is considered to be the set of all possible input values of x for which the function is defined.

So,

$\mathrm{Domain\:of\:}\:\frac{1}{3}\cdot \:27^x\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<x<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}$

As range is the set of dependent variable for which the function is defined.

As

$\mathrm{The\:range\:of\:an\:exponential\:function\:of\:the\:form}\:c\cdot \:n^{ax+b}+k\:\mathrm{is}\:\:f\left(x\right)>k$

$k=0$

$f\left(x\right)>0$

Therefore,

$\mathrm{Range\:of\:}\frac{1}{3}\cdot \:27^x:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)>0\:\\ \:\mathrm{Interval\:Notation:}&\:\left(0,\:\infty \:\right)\end{bmatrix}$

Graph is also attached from where you can observe all the key aspects which have been discussed above.

Keywords: domain, range, function, graph

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