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Kristin Fracchia

TuesdACT Video: Red Book Math Test 5 #60 – Functions


In this special episode of TuesdACT (and the next few episodes), we are going to work through solutions to some of the questions in The Real ACT Prep Guide, 3rd Edition that students are always asking about. We’re kicking it off with a math question on functions from ACT Practice Test 5, Question 60, Page 747!

Check out the video (above) for a full explanation and/or read the solution below!

Math Question 60 in Practice Test 5 is a classic example of an ACT question that looks pretty intimidating but is actually not that bad once you realize all you need to do is apply the same rules that you’ve learned in school and you’ll be ok. This is a pretty classic standardized test trick.

The question tells us that f(a,b) = b^2 - a and asks us to find what f((x^2 + y^2), (x - y)) equals.

Let’s review functions with a simpler example.

A function is like a machine that has an input and an output. So let’s say that our function is f(a,b) = b^2 - a as our problem tells us, and let’s pretend that a = 3 and b = 4.

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a=3 and b=4 is our “input” so we plug those numbers into our function machine and we get:

f(a,b) = b^2 - a
f(a,b) = 4^2 - 3 = 13

13 is our “output.:

That’s all we need to do on problem 60 on Test 5, except now we are plugging in algebraic equations.

We need to sub in x^2 + y^2 for a, and x - y for b

f((x^2 + y^2), (x - y))  = (x - y)^2  -  (x^2 + y^2)

We now have two options: We can work it out algebraically or we can plug in numbers for x and y and see which answer choice matches up.

Here’s the algebra solution:

First, FOIL the expression for a:

(x - y)^2 = x^2 - 2xy + y^2

The entire expression then is:

x^2 - 2xy + y^2 - (x^2 + y^2)

The x^2 and y^2 cancel out so we end up with -2xy, making our answer K.

You could also plug in numbers for x and y once you get the expression and see which answer choice matches up.

Join us next TuesdACT for another solution to a tricky problem from The Real ACT Prep Guide!


About Kristin Fracchia

Dr. Kristin Fracchia currently focuses on our MCAT and LSAT Prep, but she also has expertise in a wide range of standardized tests, including the ACT, SAT, GRE, and GMAT, as well as college and grad school admissions. With a PhD from UC Irvine and degrees in Education and English, she’s been working in education since 2004. She enjoys the agony and bliss of long distance trail running, backpacking, hot yoga, and esoteric knowledge.

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