Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).
( f(x) = 4x^{6}-9x^{5} )
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the domain of ( f ) is ( (-infty,infty) ).
(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)
b. the domain of ( f ) is empty.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the ( x )-intercept(s) of ( f ) is/are ( x = ).
(type an exact answer, using radicals as needed. type an integer or a fractic. use a comma to separate answers as needed.)
b. the function ( f ) has no ( x )-intercepts.

Explanation:

Step1: Find the x - intercepts

Set \(y = f(x)=0\), so \(4x^{6}-9x^{5}=0\).
Factor out \(x^{5}\): \(x^{5}(4x - 9)=0\).

Step2: Solve the equation

Using the zero - product property \(ab = 0\) implies \(a = 0\) or \(b = 0\).
If \(x^{5}=0\), then \(x = 0\).
If \(4x-9=0\), then \(4x=9\), and \(x=\frac{9}{4}\).

Answer:

A. The \(x\) - intercept(s) of \(f\) is/are \(x = 0,\frac{9}{4}\)