Calculus I: Write 120-150 words Definition paragrapghs with at least one example

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follow_this_example--domain_symmetry.pdf

Garcia & Low 1

Anabel Garcia & Judy Low

Mrs. Pirraglia

Calculus I

6 December 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 trigonometric functions), but other times

some values will need to be excluded. Finding the domain is usually 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. This is related to (but

not the same as) the range of function, which instead is the set of resulting output values.

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 so it is symmetric about the 𝑦-axis. If 𝑓(−𝑥) = −𝑓(𝑥), the graph is

odd and so it is symmetric about the origin.

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.