`f(x)=xsqrt(3-x)`   domain, end behavior, y-intercept, x-intercepts, graph it

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`f(x) = xsqrt(3-x)`

(A) Domain of the function

Note that in square roots, a negative radicand is not allowed.

So to determine the domain, consider 3-x and set it greater than and equal to zero.

`3-x gt=0`

Then, solve for x. To do so, subtract both sides by 3.



And, divide both sides by -1.

`(-x)/(-1) gt=(-3)/(-1)`

Since the x changes from negative to positive, reverse the inequality.


Hence, the domain of the given function is `(-oo, 3]` .

(B) End behavior

To determine the end behavior of the function, consider its domain.

Since the domain is `(-oo, 3]` , to get the left end behavior of the function, take the limit of f(x) as it approaches negative infinity.

`lim_(x->-oo)f(x)=lim_(x->-oo) xsqrt(3-x) = -oo*sqrt(3-(-oo))`

Note that if a number is being multiplied to a very large negative number, the product is a very large negative number.



And to get the right end behavior, take the limit of f(x) as it approaches 3.

`lim_(x->3)f(x)=lim_(x->3) xsqrt(3-x) =3sqrt(3-3)=3*0=0`

Hence, as x approaches negative infinity, f(x) approaches negative infinity. And as x approaches 3, f(x) approaches zero. Graphically, the left end of the graph goes down without bound and its right end stops at (3,0).

(C) y-intercept

To solve, set x=0. Then, plug-in this value of x to the function.

`f(x) =xsqrt(3-x)`


Hence, the y-intercept is `(0,0)` .

(D) x-intercept

To solve for x-intercept, set y=0.



Then, set each factor equal to zero.

For the first factor:

`x= 0`

And for the second factor:


Square both sides.



And, add both sides by x.



Hence, the x-intercepts are `(0,0)` and `(3,0)` .

(E) Graph.

The graph of the given function is :