*Yes. Yes, it will.*

As a GRE and SAT tutor in Boston, I find that many of my students have a hard time thinking about functions. What do they mean? What are they like? Even talking about them can be confusing: F of X this, G of Y that. Even though they are hard, students need to be familiar with functions to succeed on the SAT or the GRE. Starting with pre-calculus, math depends more and more on understanding functions. I’ve found that giving concrete examples helps students retain concepts for SAT math and GRE math tutoring. So, in this blog post, I hope to give an example that sticks in your head.

Several years ago Blendtec, a company that sold blenders, became popular for its inventive web advertisements. Traditionally, blenders are a pretty boring part of the consumer-goods market. Mostly, they just turn big chunks of food into little chunks of food. When this company started a new line of more-powerful blenders, they didn’t just say “it makes ice small in 7.5 seconds while your old blender takes 15 to do it.” Instead, they went for a more memorable approach. To advertise the power of Blendtec blenders, Blendtec started putting out a series of charming 2-minute videos where they asked “will it blend?” In early episodes, the company tried common items like marbles, hockey pucks, and pens. They all blended. Later, they blended more extreme items like lighters, plungers, or Cheez-Whiz. All liquefied. Eventually, the company started blending iPads, iPhones, and skis. The answer: Yes. No matter what you put into one of these blenders, it gets reduced to tiny little chunks.

Functions are the same way. Say f(x)=2x^{2}+x-4. If you put 3 in, you get 2(3)^{2}+(3)-4 and evaluate 18+3-4=17. If you put in 1, you would get 2(1^{2})+1-4-=2+1-4=-3. Put in x, and get out 2(x)^{2}+(x)-4. Great, it blends. Just like the blender, what you plug into any function comes out looking the same; sort-of. If you put marbles in the blender, you get marble dust. Lighters: lighter dust; iPads: iPad dust. But, like the people at Blendtec, let’s get a little fancier. Given my f(x), what is f(x-4)?

To get the answer, follow the same process and plug in (x-4) anytime you see an x. If you’re following along, you should get: 2(x-4)^{2}+(x-4)-4. Then, it’s simply a matter of applying multiplication rules. As a SAT math tutor, I always tell my students to avoid doing too much math at once. Solve the problem in little chunks. Write out 2(x^{2}-4x-4x+16)+(x-4)-4, then distribute through to get 2x^{2}-8x-8x+32+x-4-4. Once you’ve turned your knot of parentheses and coefficients into a simple chain, the expression becomes easy to evaluate: 2x^{2}-15x+24.

Seeing function as an appliance that does a specific action to whatever is “thrown in” makes learning and using functions easier. For high school students taking the SAT who want to do more math, there will only be more functions as you advance. Talk with a tutor to learn more about functions, how they work, and how frequently they appear in places you wouldn’t expect.

Now, after you’ve read all that, you deserve some fun. Go watch the video where they blend lighters. It’s…illuminating.