`f(x)=xsqrt(3-x)` domain, end behavior, y-intercept, x-intercepts, graph it
`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.
Then, solve for x. To do so, subtract both sides by 3.
And, divide both sides by -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).
To solve, set x=0. Then, plug-in this value of x to the function.
Hence, the y-intercept is `(0,0)` .
To solve for x-intercept, set y=0.
Then, set each factor equal to zero.
For the first factor:
And for the second factor:
Square both sides.
And, add both sides by x.
Hence, the x-intercepts are `(0,0)` and `(3,0)` .
The graph of the given function is :