Question

Identify the key characteristics of the parent fifth-root function f (x) = ^5sqrtx. Include the following: domain, range, intervals where the function is increasing, decreasing, or constant, intercepts, where the function is even, odd, both or neither, whether the function is continuous or discontinuous, asymptotes, and end behavior.
sqrt=square root.
The square root is supposed to be to the fifth power.

Answer 1

Given functin is :

$f\left(x\right)=\sqrt[5]{x}$

We know that the domain of the expression is all real numbers except where the expression is undefined. In given function, there is no real number that makes the expression undefined. Hence domain is all real numbers.

Domain: (-∞,∞)

Range is the set of y-values obtained by plugging values from domain so the range will also same.

Range: (-∞,∞)

If we increase value of x then y-value will also increase so that means it is an INCREASING function. You can also verify that from graph.

It crosses x and y-axes both at the origin

Hence x-intercept=0 and y-intercept=0

Graph is not symmetric about y-axis hence it can't be EVEN

Graph is not symmetric about origin so it is ODD.

There is  no breaking point in the graph so that means it is a Continuous function.

There is no hoirzontal or vertical or slant line which seems to be appearing to touch the graph at infinity so there is NO asymptote.

END behaviour means how y-changes when x approaches infinity.

From graph we can see that when x-approaches -∞ then y also approaches ∞.

when x-approaches +∞ then y also approaches +∞.