### Video Transcript

Determine the domain and the range
of the function π of π₯ equals four multiplied by π₯ minus four all squared minus
three.

Firstly, we recall that the domain
is the set of all values on which the function acts, which we can also think of as
the set of input values to the function. As the function π of π₯ is a
polynomial and, more specifically, a quadratic, there are no restrictions on what
values it can act on. Therefore, we say that the domain
of this function is the set of all real numbers.

The range of a function is the set
of all values the function produces, which we can think of as the set of all output
values. To determine the range of a
quadratic function, we can consider its turning point. Now, this quadratic function has
been given to us in its completed square or vertex form. π of π₯ equals π multiplied by π₯
plus π all squared plus π. And we know that when a quadratic
function is given in this form, its vertex has the coordinates negative π, π. The value of π for our quadratic
is negative four, and the value of π is negative three. So the vertex will be at negative
negative four, thatβs four, negative three.

As the value of π, the coefficient
of π₯ squared in our quadratic function, is four, which is positive, we know that
its graph will be a parabola which curves upwards. So this vertex of four, negative
three will be a minimum point. The possible values of π of π₯
then will be all the values from this minimum value of the function negative three
upwards.

We can express this either as π of
π₯ is greater than or equal to negative three or using interval notation as the
interval from negative three to β, which is closed at the lower end and open at the
upper end. We can answer the problem then by
saying that the domain of this function is the set of all real numbers and the range
is the interval from negative three to β, which is closed at the lower end and open
at the upper end.