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) = t^{4}

q(t) = 2t^{2}

r(t) = t^{4} − 2t^{2}

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 t^{4} is higher than t^{2}. 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 t^{4} − 2t^{2}. For small values, 2t^{2} may be greater than t^{4}. 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.

**The correct answer is (D).**

r(t) = p(t) − q(t)

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

p(t) = t

q(t) = 2t

r(t) = t

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 t

t = 0

r(0) = (0)

t = 1

r(1) = (1)

t = 2

r(2) = (2)

t = 3

r(3) = (3)

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.

## 2 Comments

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

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!