Step-by-step explanation:
Considering the function
[tex]f\left(x\right)=\:\left(\frac{1}{10}\right)^x[/tex]
Analyzing option A)
Considering the function
[tex]f\left(x\right)=\:\left(\frac{1}{10}\right)^x[/tex]
Putting [tex]x = 1[/tex] in the function
[tex]f\left(1\right)=\:\left(\frac{1}{10}\right)^1[/tex]
[tex]f\left(1\right)=\:\left\frac{1}{10}\right[/tex]
So, it is TRUE that when [tex]x = 1[/tex] then the out put will be [tex]f\left(1\right)=\:\left\frac{1}{10}\right[/tex]
Therefore, the statement that '' The graph contains [tex]\left(1,\:\frac{1}{10}\right)[/tex] '' is TRUE.
Analyzing option B)
Considering the function
[tex]f\left(x\right)=\:\left(\frac{1}{10}\right)^x[/tex]
The range of the function is the set of values of the dependent variable for which a function is defined.
[tex]\mathrm{The\:range\:of\:an\:exponential\:function\:of\:the\:form}\:c\cdot \:n^{ax+b}+k\:\mathrm{is}\:\:f\left(x\right)>k[/tex]
[tex]k=0[/tex]
[tex]f\left(x\right)>0[/tex]
Thus,
[tex]\mathrm{Range\:of\:}\left(\frac{1}{10}\right)^x:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)>0\:\\ \:\mathrm{Interval\:Notation:}&\:\left(0,\:\infty \:\right)\end{bmatrix}[/tex]
Therefore, the statement that ''The range of [tex]f(x)[/tex] is [tex]y > \frac{1}{10}[/tex] " is FALSE
Analyzing option C)
Considering the function
[tex]f\left(x\right)=\:\left(\frac{1}{10}\right)^x[/tex]
The domain of the function is the set of input values which the function is real and defined.
As the function has no undefined points nor domain constraints.
So, the domain is [tex]-\infty \:<x<\infty \:[/tex]
Thus,
[tex]\mathrm{Domain\:of\:}\:\left(\frac{1}{10}\right)^x\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<x<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}[/tex]
Therefore, the statement that ''The domain of [tex]f(x)[/tex] is [tex]x>0[/tex] '' is FALSE.
Analyzing option D)
Considering the function
[tex]f\left(x\right)=\:\left(\frac{1}{10}\right)^x[/tex]
As the base of the exponential function is less then 1.
i.e. 0 < b < 1
Thus, the function is decreasing
Also check the graph of the function below, which shows that the function is decreasing.
Therefore, the statement '' It is always increasing '' is FALSE.
Keywords: function, exponential function, increasing function, decreasing function, domain, range
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