A FUNCTION..... Finding the RANGE....?

Question

The problem is this:

For function f(x) = -3x^2 + 21x - 18

the domain is: ___________
the range is: _____________

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The domain is all real numbers, but what is the range for this problem? Please help me.

Answer 1

To find the range of a parabola, you must first convert it to vertex form; that is, convert it to this form.

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

If "a" is positive, the range will be y >= k (since the parabola will have a minimum at that point).
If "a" is negative, the range will be y <= k (since the parabola iwll have a maximum at that point).

To put your function in that form, you must complete the square.

f(x) = -3x^2 + 21x - 18

Factor (-3) out of the first two terms.

f(x) = -3(x^2 - 7x) - 18

Add "half squared" of the coefficient of x inside the brackets, and offset this by subtracting -3 times the term added outside of the brackets.

f(x) = -3(x^2 - 7x + 49/4) - 18 - (-3)(49/4)

Factor the now-square trinomial.

f(x) = -3(x - 7/2)^2 - 18 + 147/4
f(x) = -3(x - 7/2)^2 - 72/4 + 147/4
f(x) = -3(x - 7/2)^2 + 75/4

Now that this is of the vertex form, we conclude that our vertex is at (7/2, 75/4).

Since a = -3 (negative), our range is y <= 75/4.
In interval notation, (-infinity, 75/4]

Answer 2

Domain: all the values that x can take. Here, there are no restrictions, therefore x can be anything. You say all real numbers, then we continue with real numbers.

Range = all the values that f(x) can reach.
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Well, are there real numbers that cannot be reached by the function?

Because this is a quadratic, I suspect that it has a critical value (a.k.a. apex): a minimum or a maximum. In this case (because of the minus sign in front of the x^2), it is safe to assume that it can reach any large negative number (It would graph as a parabola with the 'arms' pointing down).

Does it reach zero?
Does it have a root (this is the same question)
f(x) = (-3x +18)( x-1)
Because it factors, the answer is yes.
So far, our range includes all negative numbers and zero.

Because there are two distinct roots
(+1 and +6)
f(x) will go into positive values until it reaches its maximum midway between the roots (at x= +3.5) where f(x) = +18.75.

The range is from -infinity to +18.75 (inclusive).
This can be written in the format used by pedro2008:
f(x) is equal to or less than +18.75

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Example: is there a value that we can put for x that will give us f(x) = +20 ? No. Therefore +20 is not in the range.

Can we find a value for f(x) if we put x = +20 ? yes.
f(20) = -728.
Therefore +20 is in the domain (but we knew that already).

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Domains can be tricky if you have divisions (watch for divisions by zero) or other functions that restrict the values you can use, like a square root (can't do square roots of negative numbers in real numbers) or logarithms.

PS(I stumbled on the signs the first time around -- sorry)

Answer 3

The range is all the possible Y values of the function. In this case, if you graph it, it never goes above 18.75, so the range is:

Y <= (less than or equal to)18.75

Answer 4

I FORGOT HOW TO DO THIS BUT IT THINK THE DOMAIN IS 3 AND THE RANGE IS 18