What is the domain and range of #f(x) = -7(x - 2)^2 - 9#?

1 Answer
May 5, 2018

Answer:

See below.

Explanation:

#-7(x-2)^2-9#

This is a polynomial, so its domain is all #RR#.

This can be expressed in set notation as:

#{x in RR}#

To find the range:

We notice that the function is in the form:

#color(red)(y=a(x-h)^2+k#

Where:

#bbacolor(white)(88)#is the coefficient of #x^2#.

#bbhcolor(white)(88)# is the axis of symmetry.

#bbkcolor(white)(88)# is the maximum or minimum value of the function.

Because #bba# is negative we have a parabola of the form, #nnn#.

This means #bbk# is a maximum value.

#k=-9#

Next we see what happens as #x-> +-oo#

as #x->oo# , #color(white)(8888)-7(x-2)^2-9->-oo#

as #x->-oo# , #color(white)(8888)-7(x-2)^2-9->-oo#

So we can see that the range is:

#{y in RR | -oo < y <= -9}#

The graph confirms this:

graph{-7x^2+28x-37 [-1, 3, -16.88, -1]}