what are the domain and range of the function f(x) = 4(3 square root 81)^x?{x| x is a real number}; {y| y > 0}{x| x > 4};
2 answers:
we are given
Domain:
we know that domain is all possible values of x for which any function is defined
Here , since, x is only exponent
so, we can take any values of x
it will be defined for all values of x
so, domain is all real numbers
{x | x is a real number}
Range:
we know that
range is all possible values of f(x) or y
Since, there is no negative sign here
so, f(x) will always be positive and greater than 0
so, range is y>0
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The radius of convergence R is 1 and the interval of convergence is (-3, -1) for the given power series. This can be obtained by using ratio test.
<h3>Find the radius of convergence R and the interval of convergence:</h3>Ratio test is the test that is used to find the convergence of the given power series.
First aₙ is noted and then aₙ₊₁ is noted.
For ∑ aₙ, aₙ and aₙ₊₁ is noted.
= β
- If β < 1, then the series converges
- If β > 1, then the series diverges
- If β = 1, then the series inconclusive
Here =
and
=
Now limit is taken,
=
=
=
=
= < 1
- 1 < < 1
- 1 - 2 < x < 1 - 2
- 3 < x < - 1
We get that,
interval of convergence = (-3, -1)
radius of convergence R = 1
Hence the radius of convergence R is 1 and the interval of convergence is (-3, -1) for the given power series.
Learn more about radius of convergence here:
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