Home, Home on the Range

The first week of school I gave this problem and never came back to it:

Find the range of f(x)=2^{x^2-4x+1}.

 

The answer is \left[\frac{1}{8},\infty \right). Here’s why:

First consider the range of the quadratic in the exponent, h(x)=x^2-4x+1.  It’s a parabola that opens up with its vertex at (2,-3). So the range of h(x) is [-3,\infty).

Now, consider the function g(x)=2^x.  If we let the domain of g be the values coming from  the range of h, we have the mapping \left[-3,\infty\right)\to\mathbb{R}. That is, we’re considering the composition f(x)=g(h(x)). Since g is monotonically increasing, for any x_1<x_2, we know g(x_1)<g(x_2). So the range of g in \mathbb{R} is \left[g(-3),\infty\right). So the range of f is \left[\frac{1}{8},\infty \right).

Do you feel “at home on the range”?

Here are a few more for you to try. In each case, find the range of the function. These aren’t meant to be any harder than the original problem, just different. Though watch out for the third one :-).

y=3^{-x^2+4}

y=4^{\sqrt{x}}

y=\ln{(x^2-x+1)}

y=\sin{(x^2+2x+5)}

Powerful Problem (hint)

A few weeks ago I posted this “powerful” problem:

Solve \left(x^2-5x+5\right)^{\left(x^2-9x+20\right)}=1

Now, allow me to give you a major hint. Consider the simpler equation

a^b=1

What are the possible values of a and b? Here are the possible combinations:

    • a=1 and b is anything
    • b=0 and a is any nonzero number

And here’s the tricky one that most people forget:

    • a=-1 and b is even

You now have enough information to solve the original equation. I think you’ll be delighted with the solution!

Powerful Problem

I love this problem. I love it because it seems so complicated at first, just because we don’t teach students how to attack problems like this in Algebra class. There aren’t any “traditional” methods of attacking it, just a little mathematical reasoning/logic. Here it is:

Solve \left(x^2-5x+5\right)^{\left(x^2-9x+20\right)}=1

And this is my new “super duper” problem which I post throughout the year on my board (I use a lot of the same problems each year). I first saw this problem at Messiah College where one of my professors shared it–either Dr. Phillippy or Dr. Brubaker, I can’t remember which.

So give it a try. It’s sure to delight you. My Precalculus class was sharp enough to solve it today in one period (albeit, while I was teaching about a completely different topic :-)).

Interesting Cube Problem

If the cube has a volume of 64, what is the area of the green parallelogram? (Assume points I and J are midpoints.)

Go ahead, work it out. Then, go here for a more in depth discussion, including a video explanation. Also, see my very simple solution in the comments on that page. (My Precalculus students should especially take note!)

And, welcome, SAT Math Blog, to the internet! Thanks for pointing us to this great problem and creating the nice diagram above.

Soda Mixing Problem (revisited)

I posted a problem back in December that I never got back to answering. Sorry about that. The problem statement was:

Two jars contain an equal volume of soda. One contains Sprite, the other Coca Cola. You take a small amount of Coca Cola from the Coca Cola jar and add it to the Sprite jar. After uniformly mixing this concoction, you take a small amount out and put it back in the Coca Cola jar, restoring both jars to their original volumes. After having done this, is there more Coca Cola in the Sprite jar or more Sprite in the Coca Cola jar? Or, are they equally contaminated?

I have had the worked out solution for a while, just haven’t posted it until now. I’m relatively new with \LaTeX, but I’ve typed up the solution here, if you want all the gory details :-).  And yes, Peekay, you got the right answer!