Finding Range of an Unfactorable Quartic function

[hellomyjello]

f(x) = 6x⁴+7x²-8x+9

I need to find the range of this function algebraically,

What I've done

1. Tried to factor however I can not find a way to. I've tried using online calculators and no luck either.
2. Found the Y-intercept (0,9) and tried to find the vertex (thus range), however, I do not know how to that from here.
3. Replace x⁴ with u² in order to try and make a quadratic. However the -8x doesn't let me go this route.

I can not use a graph, I know this would be the easiest way. I'm looking at how to algebraically do this.

I've tried to keep things as concise and clear as possible. This in my first post so I hope I accomplished that. Thank you for your time and help.

 

[Deleted member 4993]

f(x) = 6x⁴+7x²-8x+9

I need to find the range of this function algebraically,

What I've done

1. Tried to factor however I can not find a way to. I've tried using online calculators and no luck either.
2. Found the Y-intercept (0,9) and tried to find the vertex (thus range), however, I do not know how to that from here.
3. Replace x⁴ with u² in order to try and make a quadratic. However the -8x doesn't let me go this route.

I can not use a graph, I know this would be the easiest way. I'm looking at how to algebraically do this.

I've tried to keep things as concise and clear as possible. This in my first post so I hope I accomplished that. Thank you for your time and help.

Are you sure that the problem ask for you to find the DOMAIN of the polynomial - instead of range of the function?

If the question really asking for the range of the function you will need to calculate the maxima/minima of the function.

Do you know how to do that?

 

[tkhunny]

f(x) = 6x⁴+7x²-8x+9

Some reasoning...

Even degree, Positive first coefficient -- This thing has a minimum value.

The ONLY thing forcing this function downward, is that "-8x" for x > 0.

Some hard work...

Our outnumbered friend "-8x" will be massively overpowered if we wander too far from zero (0). Let's just try x = 0, 1
f(0) = 9
f(1) = 14
Note: I did NOT try x = -1, because those even exponents won't give us any help with negative numbers and we already established that "-8x" helps only for positive values of x.

Warning: Keep your eyes on the denominators of the input values. I deliberately wrote the input values as fractions to emphasize the nature of the process.

Let's poke around between x = 0 and x = 1.
f(1/2) = 7.125 -- Nice, even lower.

What shall we try next?
f(3/4) = 8.836 -- Well, that's probably a loser.
f(1/4) = 7.461 -- That didn't help much, either.

Poke around a little more
f(5/8) = 7.650 -- Well, that's probably a loser.
f(3/8) = 7.103 -- We may have a winner.

Let's stop looking at x > 1/2
f(7/16) = 7.060 -- Current record holder. Can you improve it?

Note: This is just one way to go about it. It's a rough example of something called "Bisection". It really isn't all that convenient or all that fast, but it is systematic. A spreadsheet might help you on your way a little faster.

 

[hellomyjello]

The trial and error method is a great idea, thank you for it. I'll work on going through that process in a timely fashion as I need to be able to use it during a timed exam. I really like the reasoning behind it.

As for finding the maxima and minima, I'm going to learn how to do so and update the tread. It looks promising. Thanks a lot for your guy's help.