assignment

Garcia & Low 1

Anabel Garcia & Judy Low

Mrs. Pirraglia

Calculus I

4 May 2015

Writing to Define

DOMAIN

The domain of a function is a set that represents all possible input values. Sometimes the

domain will be all real numbers (like for polynomial and trig functions), but other times some

values will need to be excluded. Most of the time, looking for the domain is easier when we try to

figure out if any numbers cause a problem for the function. It’s like being lactose intolerant and

asking your doctor about a good diet. Instead of naming every food item that is okay, it makes

more sense for the doctor to identify the few things that will make you sorry if you do eat them.

Some examples:

𝑓(𝑥) = 𝑥4 − 3𝑥2 + 9𝑥 − 1

The domain of a polynomial is all real numbers.

𝑔(𝑥) = 𝑥5 − 11𝑥 + 4

(𝑥 − 3)(2𝑥 − 1)

The domain is {𝑥|𝑥 ≠ 3, 𝑥 ≠ 1

2 }. The denominator equals zero for these 𝑥-values so the rational

function 𝑔 is undefined there.

ℎ(𝑥) = log3 𝑥

The domain is {𝑥|𝑥 > 0}. Since log and exponential functions are inverses, it makes sense that if

the output of an exponential function is always positive, then the input of the logarithmic

function must also always be positive.

Garcia & Low 2

SYMMETRY

The symmetry of a function is a property which describes how the graph may be reflected

onto itself. An even function is symmetric about the 𝑦-axis, so the 𝑦-axis acts like a mirror where

the left side looks like a reflection of the right side and vice versa. Also, if the paper is folded

along the 𝑦-axis, the left and right sides will meet. An odd function is symmetric about the

origin, and this graph looks the same if you spin the paper around 180o and look at it upside

down. Even though symmetry describes what the graph looks like, we use algebra to test for this

property. If 𝑓(𝑥) = 𝑓(−𝑥), the graph is even, and if 𝑓(−𝑥) = −𝑓(𝑥), the graph is odd.

Some examples:

𝑓(𝑥) = 𝑥6 − 3𝑥4 + 12

Since 𝑓(−𝑥) = (−𝑥)6 − 3(−𝑥)4 + 12 = 𝑥6 − 3𝑥4 + 12 = 𝑓(𝑥), the function

is even. While not all even functions are polynomial or even rational functions,

notice that in this polynomial all the exponents are even.

𝑔(𝑥) = 𝑥6 + 2

8𝑥3

Here, 𝑔(−𝑥) = (−𝑥)6+2

8(−𝑥)3 =

𝑥6+2

8(−𝑥3) = −

𝑥6+2

8𝑥3 = −𝑔(𝑥), so the function is odd. Note that we can

rewrite 𝑔(𝑥) as 𝑔(𝑥) = 𝑥6

8𝑥3 +

2

8𝑥3 =

1

8 𝑥3 +

1

4𝑥3 so that we see the odd

powers more easily.