# Let’s Get Functional – SAT Math Question of the Day

Explanation
In this question, we need to do operations between functions. Three functions p, q and r are described depending on the variable t. Function r is defined in terms of functions p and q as follows:

r(t) = p(t) − q(t)

We can plug the values of p(t) and q(t) into the equation above:

p(t) = t4
q(t) = 2t2
r(t) = t4 − 2t2

For t > 0, we are asked how r changes with increasing t (t gets farther from 0 means it increases in value). The increase ratio of t4 is higher than t2. This means that the value of r increases. However, there is a point we need to be careful about. At the beginning, the function characteristic may be different due to t4 − 2t2. For small values, 2t2 may be greater than t4. It is better to examine the function by assigning numbers for t:
t = 0
r(0) = (0)4 − 2(0)2 = 0
t = 1
r(1) = (1)4 − 2(1)2 = −1
t = 2
r(2) = (2)4 − 2(2)2 = 8
t = 3
r(3) = (3)4 − 2(3)2 = 63

We see that in the interval 0 < t < 1, r(t) first decreases from 0 to -1, and then increases from -1 to 8 and continues increasing as t gets farther from 0.

1. Mohammed says:

as someone used to long procedures,i found this app very interesting especially the shortcuts used to solve the problems. I only had one problem with the question on July 21,2015. I still think my answer (A) is correct because in the question, t is said to be more than zero. I need your feedback please. thank u

• LTG Exam Team says:

Hey Mohammed,

Yes, you’re right: t is said to be greater than zero, so using t = 0 as a starting point doesn’t appear correct. However, the correct answer is still D. Take for instance if t = .5 .

.5 is greater than zero, and when t = .5, r is -.4375.
Then when t is 1, r is -1.
And when t is 2, r is 8.
And when t is 3, r is 63 and so forth.

The change from -.4375 to -1 is a decrease because -1 is further away from zero than -.4375 .

So as t increases from above zero (for example, imagine the smallest possible decibel above zero), r initially decreases and then increases.

Hope this helps!