Question

How do you find the domain of \[f\left( x \right)=-{{x}^{2}}\]

Answer 1

Hint: to get the domain first find out the nature of the equation (parabola), then find out the vertex, focus and directrix of the given equation then plot the desired graph of the equation.
Points can be taken to plot the graph and then the domain can be easily found out by the graph.

Complete step by step solution:
domain is designed to insure that function is well defined.
That gives the acceptable inputs.
To find the domain we have to draw the graph first, we can see that it’s the equation of a parabola.
Now according to the equation of parabola:
\[\begin{align}
  & {{x}^{2}}=-4ay \\
 & a>0 \\
\end{align}\]
Comparing the standard equation to the given equation,
Here \[a\] is equal to \[\dfrac{1}{4}\]
And,
\[f\left( x \right)=y\]
Vertex of the parabola is \[(0,0)\]
Focus of the parabola is \[(0,-1/4)\]
Equation of the directrix is \[y=1/4\]
Equation of axis is \[x=0\]
To draw the graph of the given parabola we take some points,
Putting values of \[x\] to get \[y\];
\[\begin{align}
  & x=-2,y=-4 \\
 & x=-1,y=-1 \\
 & x=0,y=0 \\
 & x=1,y=-1 \\
 & x=2,y=-4 \\
\end{align}\]
Using these points we plot the graph as:

From the given graph the domain is the input that enable the function so here the domain is all numbers
\[domf-\left( -\infty ,\infty \right)\]
That belongs to all real numbers as the graph goes endlessly to both the sides of the axis.

Note: we cannot divide it by zero.
We cannot take the square root of negative number
Square root cannot be negative.
\[f(x)=\dfrac{1}{g(x)}\ne 0\]
Only acceptable inputs are allowed to get the desired domain .