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

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Answer 1 · 2018-05-05T15:39:22

See below.

$- 7 {(x - 2)}^{2} - 9$ $-7(x-2)^2-9$

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]}

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