Question

The graph of f(x) = StartRoot x EndRoot is reflected across the x-axis and then across the y-axis to create the graph of function g(x). Which statements about the functions f(x) and g(x) are true? Check all that apply.
The functions have the same range.
The functions have the same domains.
The only value that is in the domains of both functions is 0.
There are no values that are in the ranges of both functions.
The domain of g(x) is all values greater than or equal to 0.
The range of g(x) is all values less than or equal to 0.

Answer 1

Step-by-step explanation:

Considering the function

$f(x)=\sqrt{x}$

As domain is the set of all the input values for which function is defined and range is the set of all output values which are obtained after we have substituted the values in the domain.

So,

$\mathrm{Domain\:of\:}\:\sqrt{x}\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x\ge \:0\:\\ \:\mathrm{Interval\:Notation:}&\:[0,\:\infty \:)\end{bmatrix}$

and

$\mathrm{Range\:of\:}\sqrt{x}:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)\ge \:0\:\\ \:\mathrm{Interval\:Notation:}&\:[0,\:\infty \:)\end{bmatrix}$

After the first transformation of $f(x)=\sqrt{x}$, i.e. the reflection of f(x) over the x axis, the new image will be

$f\left(x\right)=-\sqrt{x}$

After, the second transformation, the reflection of $f\left(x\right)=-\sqrt{x}$ over the y-axis will bring final image as $g\left(x\right)=-\sqrt{-x}$

For $g\left(x\right)=-\sqrt{-x}$

$\mathrm{Domain\:of\:}\:-\sqrt{-x}\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:x\le \:0\:\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:0]\end{bmatrix}$

As

$\mathrm{The\:range\:of\:an\:radical\:function\:of\:the\:form}\:-c\sqrt{ax+b}+k\:\mathrm{is}\:\:f\left(x\right)\le \:k$

$k=0$

$f\left(x\right)\le \:0$

So,

$\mathrm{Range\:of\:}-\sqrt{-x}:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)\le \:0\:\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:0]\end{bmatrix}$

Checking the statements:

A) The functions have the same range is FALSE as the range changed from $y\:\ge \:0\:$  to $y\:\le \:0$

B)The functions have the same domains is also FALSE as the domain changed from $x\:\ge \:0$ to $x\:\le \:0$.

C)The only value that is in the domains of both functions is 0 is TRUE as the intersection of $x\:\ge \:0$ with  $x\:\le \:0$ is $0$.

D)There are no values that are in the ranges of both functions is FALSE as $0$ is in the ranges of both functions.

E)The domain of g(x) is all values greater than or equal to 0 is FALSE as it is clear that the domain of $g(x)$ is all values less than or equal to $0$.

F)The range of g(x) is all values less than or equal to 0 is TRUE as it was observed in our calculations.

Keywords: graph, radical expression, domain, range

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Answer 2

Answer:

C and F

Step-by-step explanation:

E2020