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

Answer:

The only value that is in the domains of both functions is 0

The range of g(x) is all values less than or equal to 0

Step-by-step explanation:

As the original function is

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

Since, domain is the set of all possible input values that define the function, and range is the set of all possible output values for all possible domain values for which the function is defined.

• The domain of $f(x) = \sqrt{x}$ will be [0, ∞)

• The range of $f(x) = \sqrt{x}$ will be [0, ∞)

Please check the attached figure a for visualizing the graph of $f(x) = \sqrt{x}$.

Impact of double transformation:

• When the function $f(x) = \sqrt{x}$ is reflected across x-axis, the function becomes $y = -\sqrt{x}$ after first transformation

• After the second transformation across y-axis, the function $y = -\sqrt{x}$ becomes  $g(x) = -\sqrt{-x}$

For

$g(x) = -\sqrt{-x}$

$-x$ must be equal to or greater than zero for $g(x) = -\sqrt{-x}$ to be defined i.e. -x ≥ 0.

So,

-x ≥ 0 can be written as x≤ 0

So,

• The domain of $g(x) = -\sqrt{-x}$ will be (∞, 0]
• The range of $g(x) = -\sqrt{-x}$ will be (∞, 0]

Please check the attached figure a for visualizing the graph of $g(x) = -\sqrt{-x}$.

So, from the above discussion, we can say that

• 0 is the only that is in the domain of both function.
• The range of g(x) is all values less than or equal to 0

So,

Only two statements are true about the functions f(x) and g(x) are true which are:

The only value that is in the domains of both functions is 0

The range of g(x) is all values less than or equal to 0

Keywords: graph, function

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Answer 2

Answer:

3rd and last option

Step-by-step explanation:

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