It is not necessary true that $\displaystyle f(g(x)) = x \implies g(f(x)) = x$. I remember that CaptainBlank once gave an example of that.
Lesson 1.3 - Functions: Even & Odd Functions
The function $\displaystyle y = f(x)$ is even if $\displaystyle f(-x) = f(x)$.
Even functions are symmetric about the y-axis.
The function $\displaystyle y = f(x)$ is odd if $\displaystyle f(-x) = -f(x)$.
Odd functions are symmetric about the origin (the line $\displaystyle y=x$)
They are named for the power functions which satisfy each condition: the function $\displaystyle x^n$ is:
• an even function if n is an even integer
• an odd function if n is an odd integer
Figure 1.3-A
$\displaystyle f(x) = x^2$, an example of an even function.
Notice that $\displaystyle f(x) = f(-x)$, so $\displaystyle f(3) = 9$ & $\displaystyle f(-3) = 9$
In Figure 1.3-A, any value on the right side (1st quadrant) is mirrored to the left side (2nd quadrant)
Other examples of even functions are $\displaystyle |x|$, $\displaystyle x^2$, $\displaystyle x^4$, and $\displaystyle cos(x)$
Figure 1.3-B
$\displaystyle f(x) = x^3$, an example of an odd function.
Notice that $\displaystyle f(x) = -f(x)$, so $\displaystyle f(3) = 9$ & $\displaystyle f(-3) = -9$
In Figure 1.3-B, any value in the 1st quadrant is reflected across the y-axis and once more across the x-axis into the 3rd quadrant. The same is true for the reflection from the 4th to 2nd quadrants.
Other examples of odd functions are $\displaystyle x$, $\displaystyle x^3$, and $\displaystyle sin(x)$
Figure 1.3-C
$\displaystyle f(x) = x^3 + 1$ is neither even nor odd as shown in Figure 1.3-C.
The only function which is both even and odd is: $\displaystyle f(x) = 0$.
Combining even and odd functions can provide different results. Even and odd functions follow the same properties that any function does as was shown in Lesson 1 (Functions – Basic Properties).
Do not worry about remembering these following properties of even and odd functions. The best way to see if a combination of two functions is even, odd, or neither is to graph it!
• Sum of two even functions is even, and any constant multiple of an even function is even.
• Sum of two odd functions is odd, and any constant multiple of an odd function is odd.
• Product of two even functions is an even function.
• Product of two odd functions is again an even function.
• Product of an even function and an odd function is an odd function.
• Quotient of two even functions is an even function.
• Quotient of two odd functions is an even function.
• Quotient of an even function and an odd function is an odd function.
• Composition of two odd functions is odd.
• Composition of any function with an even function is even.
Let’s try a graphical example of a sum of two even functions.
Example 1.3-A
Let $\displaystyle f(x) = x^2$, $\displaystyle g(x) = cos(x)$
$\displaystyle h(x) = f(x) + g(x)$
Both $\displaystyle f(x)$ and $\displaystyle g(x)$ are even functions, so $\displaystyle h(x)$ should be an even function as well. The right side of the y-axis should be mirrored on to the left side.
The graph in Figure 1.3-D shows that $\displaystyle h(x)$ is indeed an even function.
Figure 1.3-D
Nice tutorial.
I never realize 0 was the only function that is both even and odd until now, never really thought about it. Hope it comes in handy some day
i guess you could have clarified what you meant by symmetric in the line y = x. but i guess your graphs clear it up. It's not really a symmetry, but a kind of reverse symmetry if you look at the graphs. it's like you reflect in y = x, and then turn the reflection upside down, or something...
Thanks for the constant critiquing Jhevon. I could rewrite that line. It is symmetric about the origin (0, 0). To be symmetric about the origin, it must both be symmetric about the x-axis and the y-axis. I also stated it was symmetric about the line, $\displaystyle y=x$ in parenthesis. Any other idea how I could clarify this?
Lesson 1.4 - Functions: Periodic (Trigonometric)
For Calculus I, you should have not only studied algebra before, but also trigonometry. You should be familiar with the definitions and graphs of these trigonometric (trig) functions:
• Sine
• Cosine
• Tangent
• Cotangent
• Secant
• Cosecant
Start from the middle of that list for easy relations…
$\displaystyle cot(x)=\frac{1}{tan(x)}$
$\displaystyle sec(x)=\frac{1}{cos(x)}$
$\displaystyle csc(x)=\frac{1}{sin(x)}$
The functions are called periodic because they have repeating patterns every period (or sub-domain). For example, the graph of $\displaystyle sin(x)$ in Figure 1.4-A has a period of $\displaystyle 2\pi$.
Figure 1.4-A
The following trig relations will be very helpful when encountering multiply trig functions…
$\displaystyle sin^2(x) + cos^2(x) = 1$
$\displaystyle sec^2(x) - tan^2(x) = 1$
$\displaystyle csc^2(x) - cot^2(x) = 1$
$\displaystyle sin(2x) = 2cos(x)sin(x)$
TIP: Convert all trig functions to sines and cosines for easy simplification!
Lesson 1.5 - Functions: Zeros of a Function
These occur where the function $\displaystyle f(x)$ crosses or hits the x-axis (has a value of zero). These points are also called the roots of a function. To find such points, set $\displaystyle f(x) = 0$ and solve for all values of x.
Example 1.5-A:
The zeros of $\displaystyle f(x) = x^3 - 6x^2 + 9x$ are 0 and 3…
$\displaystyle f(x) = x(x^2 - 6x + 9) = 0$
$\displaystyle = x(x-3)^2 = 0$
$\displaystyle x = 0, 3$
Figure 1.5-A
Notice how the graph of $\displaystyle f(x) = x^3 - 6x^2 + 9x$ in Figure 1.5-A crosses the x-axis at $\displaystyle (0,0)$ and hits the x-axis at $\displaystyle (3,0)$. The function has values of zero at both of these points.
Basic Rule of Thumb: A polynomial function of degree n has n zeros, provided multiple zeros are counted more than once and provided complex zeros are counted.
Keep in mind that any complex zeros of a function are not considered to be part of the domain of the function, since only real numbers domains are being considered. It is only for the purpose of counting zeros that we consider complex and multiple zeros. Complex roots come in pairs. This can be seen from the quadratic formula.
$\displaystyle ax^2+bx+c=0$
$\displaystyle x=\frac{-b \pm \sqrt{b^2 -4ac}}{2a}$
$\displaystyle x$ is a complex root if the discriminant ($\displaystyle b^2 -4ac$) is not equal to zero. Therefore, complex roots will always come in a pairs because of the $\displaystyle \pm$ sign in the numerator. Polynomials of higher powers have similar results.
There is also the Rational Zeros Theorem which is too cumbersome for use in most Calculus courses. However, it could be helpful for some students, so here it is: If $\displaystyle P(x)$ is a polynomial with integer coefficients and if $\displaystyle \frac{P}{Q}$ is a zero of $\displaystyle P(x) (P(\frac{P}{Q}) = 0)$ , then $\displaystyle p$ is a factor of the constant term of $\displaystyle P(x)$ and $\displaystyle q$ is a factor of the leading coefficient of $\displaystyle P(x)$.
There are a few steps you can take that makes this theorem easy for polynomials greater than degree of two. Arrange the terms of the polynomial in descending order by degree. Write down all factors of the constant term. Write down all the factors of the leading coefficient (coefficient of the term that has the highest degree). Divide the factors of the constant term by the factors of the leading coefficient, and that is the list of all of the possible roots.
Example 1.5-B
$\displaystyle 2x^4+3x^3-9x^2+8x+9=0$
Factors of constant: $\displaystyle 9 \rightarrow 1,3,9$
Factors of leading coefficient: $\displaystyle 2 \rightarrow 1,2$
List of possible roots: $\displaystyle \frac{1}{1}, \frac{3}{1}, \frac{9}{1}, \frac{1}{2}, \frac{3}{2}, \frac{9}{2}$.
Now, let us go through some standard examples to give you a better understanding of how to find zeros.
Example 1.5-C
$\displaystyle f(x) = sin(x), 0 \le x \le 2\pi$
$\displaystyle sin(x) = 0$
Using the unit circle as shown in Lesson 1.4, sine is zero at the horizontal axis, namely when
$\displaystyle x = 0, \pi, 2\pi$
Example 1.5-D
$\displaystyle f(x)=\frac{(x^2+4x+3)}{(x+1)}$
$\displaystyle f(x)=\frac{((x+3)(x+1))}{(x+1)}$
$\displaystyle f(x)=x+3=0$
$\displaystyle x=-3$
Even though $\displaystyle x = -1$ makes the numerator zero, it also makes the denominator zero which makes it undefined at $\displaystyle x = -1$. In the following graph, the function acts as a line and only crosses the x-axis at $\displaystyle x = -3$. There is actually a hole in the line at $\displaystyle f(-1)$.
Example 1.5-E
$\displaystyle f(x)=e^{2x}$
$\displaystyle e^{2x}=0$
$\displaystyle 2x=ln|0|\Rightarrow $undefined
It is undefined because $\displaystyle e^x>0$. Try thinking about an exponent that would make a number equal to zero. There is none, and there is no such power that will make change its sign.
Function has no zeros as evidenced by the graph in Figure 1.5-B.
Figure 1.5-B
The exponential function gets very close to zero but never actually has a value where $\displaystyle f(x) = 0$.
i didn't spot any typos, but this is no where near the amount of stuff you need to know for trig. i believe section 1.4 is incomplete, unless you plan on coming back to this stuff soon. also, giving the formula for sin(2x) without giving the one for cos(2x) or tan(2x) is awkward. at least beef up this section with a more hefty set of formulas. the addition formulas, half angle formulas, and maybe the product to sum formulas among other things would be a great addition to this section. and of course, you can't talk about sine cosine and tangent without talking about things like SOHCAHTOA
i also recommend you talk about special angles and the trig functions of special angles. a talk on reference angles would also be in order. of course, it's a lot of work, but this is a tutorial. you don't want anyone saying, "i read Colby's tutorial, but he left so many things unanswered"
i like the example you chose. to show when the function crosses the x-axis as opposed to just touching it. makes you aware of things like roots with multiplicity (that's a term you should use since texts books do it. it would make your students familiar with the vocabulary of the game)
you could also explain that complex roots come in pairs, and why they do. but maybe i'm being to picky. it might help with some problems that i see on the forum from time to time thoughBasic Rule of Thumb: A polynomial function of degree n has n zeros, provided multiple zeros are counted more than once and provided complex zeros are counted.
Keep in mind that any complex zeros of a function are not considered to be part of the domain of the function, since only real numbers domains are being considered. It is only for the purpose of counting zeros that we consider complex and multiple zeros.
you could still do a short thing on this if you have time. just state the theorem and show one example or so. would the remainder theorem and factor theorem be too much?There is also the Rational Zeros Theorem which is too cumbersome for use in most Calculus courses.
you could explain that you got the values of x from the graph of sine, or however you got it. sure, the answer seems trivial to a lot of us on the forum, but it might not be to a beginner. my point is, try not to be like a text book where it just shows you the working but doesn't explain what it's doing. make sure your students have the facilities to follow what you're doing. especially since you have such a skimpy trig sectionLet us go through some examples to give you a better understanding of how to find zeros.
Example 1.5-B
$\displaystyle f(x) = sin(x), 0 \le x \le 2\pi$
$\displaystyle sin(x) = 0$
$\displaystyle x = 0, \pi, 2\pi$
i love how you chose this example as well. to show that we have a problem if the zeros of the numerator of a fraction coincides with the zeros of the denominator. many students would just set the numerator to zero and solve, without checking if there's a problem.Example 1.5-C
$\displaystyle f(x)=\frac{(x^2+4x+3)}{(x+1)}$
$\displaystyle f(x)=\frac{((x+3)(x+1))}{(x+1)}$
$\displaystyle f(x)=x+3=0$
$\displaystyle x=-3$
Even though $\displaystyle x = -1$ makes the numerator zero, it also makes the denominator zero which makes it undefined at $\displaystyle x = -1$. In the following graph, the function acts as a line and only crosses the x-axis at $\displaystyle x = -3$. There is actually a hole in the line at $\displaystyle f(-1)$.
i think a discussion on logarithms is in order. sure the graph shows it, but where did that graph come from? why is ln|0| undefined?Example 1.5-D
$\displaystyle f(x)=e^{2x}$
$\displaystyle e^{2x}=0$
$\displaystyle 2x=ln|0|\Rightarrow $undefined
Function has no zeros as evidenced by the graph in Figure 1.5-B.
Figure 1.5-B
The exponential function gets very close to zero but never actually has a value where $\displaystyle f(x) = 0$.
General comments: as i said, i think you're doing a good job here. however it seems sometimes that you're too smart for your own good. remember that you are writing for beginners, things will not be as clear to them as it is to you. be sure to examine your work and ask yourself, "did i leave any room for doubt and/or confusion here? do i expect someone who is learning calculus to know ln(0) is undefined?" sure, there are some things you expect students studying calculus to know, but there are times when they don't remember something. the least you could do is suggest that they look it up or provide a link to a website. which brings me to a suggestion. as i said about doing a lot of extra work, you know about the rational roots theorem and all that stuff. if you don't have the time or inclination to do it, you can find a site that explains it very well, and post the link in your tutorial. make sure the link is explicit, in case someone wants to print a hard copy of the tutorial. and spend more time on trig. it's usually a very difficult part of the calculus syllabus for students, yet it's the shortest post you made. doesn't add up
indeed! this caught me in an exam once (i wrote x when i should have wrote |x|) and ever since then, the fact that $\displaystyle \sqrt{x^2} = |x|$ has been burned in my mind and i use all the time. now i have a knee jerk reaction every time i see it. i start worrying (amazingly a lot) whether the x's i'm considering are greater than or equal to zero. i never thought about it in this light though. this is an example i should have come up with
Jhevon, your recent critiques were excellent. I didn't know how to go about the trig section because students who take Calculus are to already have a good background in Trigonometry. I am sure I can add a few formulas, mention SOH-CAH-TOA, and reference to the unit circle (and All Students Take Calculus shortcut).