Domain and Range

**pashah** · August 2nd 2006, 07:53 PM

Find the domain and range for each quadratic function.

a. y = x^2 – 6x +5

b. y = 3x^2 + 2x +9

**CaptainBlack** · August 2nd 2006, 10:29 PM

Originally Posted by pashah

Find the domain and range for each quadratic function.

a. y = x^2 – 6x +5

b. y = 3x^2 + 2x +9

The RHS of both of these are defined for all of $\mathbb{R}$ , so
their domains are both the set of all real numbers.

Because these both represent parabolas which go to $\infty$ as $x$
goes to $\pm \infty$ , they can take all values greater than teir minimums.

The minimum of the first occurs when $2x-6=0$ , which is at
$x=3$ , where $y=-4$ , so for a. the range is $y \ge 4$ .

The minimum of the second occurs when $6x+2=0$ , which is at
$x=-1/3$ , where $y=26/3$ , so for a. the range is $y \ge 26/3$ .

RonL

**ThePerfectHacker** · August 3rd 2006, 07:30 AM

Originally Posted by pashah

Find the domain and range for each quadratic function.

a. y = x^2 – 6x +5

I do this slight diffrently.
Rewrite as,
$x^2-6x+(5-y)=0$
Now, you need to find all $y$ such as this quadradic equation has a real solution. That happens when, the discrimanant is non-negative.
$(-6)^2-4(1)(5-y)\geq 0$
Simplify,
$16+4y\geq 0$
Thus,
$4y\geq -16$
Thus,
$y\geq -4$

b. y = 3x^2 + 2x +9