These are my complete notes for Precalculus, covering such topics as Piecewise Functions, Composite Functions, Complex Fractions, Inverse & Parent Functions (in greater detail), Logarithms, Degrees and Radians, Trigonometry, Sinusoidal Waves, and more.
I color-coded my notes according to their meaning - All numbered notes (which I call rules) are red, and include examples and the basis for understanding a topic. Definitions are written in green, and other important information (such as large-scale drawings that are better visualized than explained) was written in blue. All of this information is preserved on this page, with logical flow and breaks. I use ascii line drawings sparingly - If I can convey information or a graph using an image online, I will do so.
All of the knowledge present in these notes are filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Precalculus (Complete)
I. Rational Equations.
Cross Multiply: Multiplying two fractions diagonally - can be done when two fractions with variables are equivalent.
Least Common Multiple: The smallest number that two or more numbers can divide into evenly.
Rule 78. Cross Multiplying is to be done in situations where there are two (2) fractions that are equal, such as (3 / (x + 1)) = (9 / (4x + 5)). This only works when both variables are by themselves.
Rule 79. Another means of finding whether an equation was extraneous or not in the fastest possible way - a rule of thumb is that if you plug in an answer you got for x in the denominator and it does not go to zero (illegal), then it is very likely to be the answer. Only use this trick in a pinch!
Rule 80. One of the more convenient aspects of the least common multiple is that, in general, it removes the denominator due to the cancelling outs. However, the goal is always to have it so that the denominator for all variables is the same, whether that be 1 or some LCM frankenstein. Always eyball beforehand whether cancelling out will grant an opportunity where all denominators are equal. For example, in the post-LCM equation (20x / x) + (28x / 4) = (-36x / x), the option is available to simplify each equation and therefore to remove all denominators. What out! It is your decision whether to multiply the denominator or not.
Rule 81. When you have gotten all variables to agree on a denominator other than 1., remember that you must simply sum the top of its plus & minuses if you want to cancel out more. For example, in the equation (5x + 3(x + 3)(x + 4)) / (6x(x + 3)), (x + 3) cannot be canceled out because there is an addition ongoing in the numerator.
Rule 82. All inequalities in regard to x (representative of the domain), such as x ≤ 12 or -4 ≤ x < -1, can be represented through interval notation. Simply, the postion of the number in the coordinates represents where the x is located between. For example, -4 < x < 2 would be (-4, 2). If an equation contains an x that goes in one direction infinitely, such as x > 12, than the infinity is used: (12, ∞). Infinity is always done with a parenthesis, as it cannot be contained. The first variable in the coordinates is the lowest possible value that x can be, naturally, and vice versa goes for the second variable. For greater or equal to symbols in the equation, whichever side has the ≤ (first or second) has a bracket used for it instead of a parenthesis. For example, -8 < x ≤ 20 = (-8, 20]. Remember that infinity can never have brackets. on a number line, a closed circle means not equal to, while a filled one means it is. For equations using the "or", a special character is used to denote these separate possibilities and limits: ∪. The Union symbol. For example, in the equation x ≥ 9 or x < 2, the coordinates would be rendered as (-∞, -2) ∪ [9, ∞).
Rule 83. As is known, a function is that which passes the vertical line test. If a vertical line intersects more than one point on the graph, it is not a function. For example, on a number chart for coordinates, X-values cannot repeat, while y values can repeat and still be a function.
Rule 84. For equations where there is an equation to solve regarding y and x, such as y² - 2x = 5, simplify as much as you can for either one and then plug in 1 for the unsimplified side's variable, in this case x: y = ±√2x + 5 = ±√7. Because the value of y is ±, there are two different values of y for one (1) x, making the equation not a function. Simply put, a ± in the value of the y makes it not a function.
Rule 85. Believe in yourself! :3
Rule 86. There are special rules for more abstract equations regarding concepts like square roots that require advanced levels of thinking. For example, in the equation g(t) = √t - 5, you must know that t - 5 ≥ 0 because a negative has no real square root. Therefore, the Domain of this equation is all real numbers that are ≥ 5: [5, ∞). Be creative! For equations that don't involve radicals, such as 3y = 11 - 4x, it is pretty much always a safe bet that the equation is a function. Squares and other regular exponents are also pretty much fine.
Rule 87. For f(x)-type equations where you plug in things for x and whatnot, more advanced and spectacular variables can be plugged in for x, just for fun. g(5x + 4) = (5c + 4)² + 8(5c + 4) - 24. Once simplified, you will find 25c² + 80c + 24, totally meaningless.
Rule 88. Another type of equation where there are Domain specificities are those where it is a fraction with a variable involving x on the denominator, such as f(x) = (2 + x) / (x² + 7x). There is nothing really to do in this equation other than finding the values that would make the denominator equal to zero, a fun game. The first step would be to simplify the denominator as much as possible, GCF and all that: x² + 7x = x(x + 7). Off the bat, you can easily tell x can't be 0, and two seconds later that x can't be -7. Therefore, the Domain is all real numbers except for x = 0 & x = 7.
Rule 89. There is a special method for plugging in x into an equation. The best way to think about it is as a computer code conditional if statement. This is a piecewise:
| 1.6x - 41.6, if 63 < x < 66
h(x) = { 3x - 132, if 66 ≤ x ≤ 68
| 2x - 66, if x > 68
Only one equation belongs to each value of x. You just go ahead with a regular 'plug-in-for-x' type problem once you know what x is. There are no special rules, it is that simple.
Rule 90. As is known of Domain and Range, the Domain is all the possible x-values a function covers while Range is all the possible y-values. Specifically, when combining Domain and Range with interval notation, Domain goes Left to Right while Range goes Down to Up. When you have a connected line graph that has no end, both ends stretching into infinity, both the domain and range are (-∞, ∞). This is for graphs that show as follows, with no asymptotes:
There can be graphs that can show disconnected lines, where functions like so:
show a puzzle to figure out what is Domain and Range, while other graphs can also use tons of dots at important points to represent whether a high/low point is included in the line. For example, in the given graph, because the highest point of the left part is an empty circle, then that point unincluded in the leftmost interval. The full interval notation would be (-8, -3) ∪ [-1, 5) ∪ (5, 8].
Rule 91. When you have a disconnected line graph, you may see that there is either a vertical or horizontal flat line. If the flat follows its respective register, such as a flat line following the x-axis when you're writing domain, no problem. But when you come across the same line while writing range, there is a special way of writing it: For a graph like this -
the first part of the interval notation would be (-∞, -2) ∪, with the down-to-up rule and all, but the second part would be represented as [6]. For either Domain or Range, brackets or parenthesis on a single digit means flat.
Rule 92. The y-intercept is found by plugging in 0 for x, f(0). The x-intercept is found by plugging in 0 for y, "0 = ax² + bx + c". X-intercept are also known as the "zeroes" and "roots" of the function.
Rule 93. There is symmetry test that can be conducted for every axis, x, y, and origin.
In relation to the x-axis:
In relation to the y-axis:
In relation to the origin:
For the origin, fold first on x and then on y. To discover algebraically if it is symmetrical on any axis, you must plug in a negative for whichever vairable you feel the equation may be symmetrical for:
X-Axis: (x, y) = (x, -y)
Y-Axis: (x, y) = (-x, y)
Origin: (x, y) = (-x, -y)
In general, for equations like x = y² - 3 or whatnot, the variable that has the square is immune to becoming negative because of the automatic squaring, so it is highly likely in said situations that a shortcut can be taken to find what axis the equation is symmetrical on. However, it is always safer to plug in the negative for each possibility to discern whether the less obvious possibility is symmetrical as well. Multiple tests should be performed if you are not sure, in order to make sure that your first test was not just a coincidence.
