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urp:algebra [2021-11-03] nerf_herder [Functions] |
urp:algebra [2021-11-08] nerf_herder |
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* [[poly|Polynomials, quadratic formula and completing the square]] | * [[poly|Polynomials, quadratic formula and completing the square]] | ||
- | **quadratic formula**: x = (-b +- √(b²-4ac)) / 2a | + | **quadratic formula**: x = (-b ±√(b²-4ac)) / 2a |
+ | |||
+ | **PEMDAS/BODMAS ** - order of operations: Parentheses/Brackets, Exponents/Order, Multiply-Divide, Add-Subtract | ||
==== System of equations==== | ==== System of equations==== | ||
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(nᵃ)ᵇ = nᵃᵇ | (nᵃ)ᵇ = nᵃᵇ | ||
n⁻ᵃ = 1/nᵃ | n⁻ᵃ = 1/nᵃ | ||
+ | |||
+ | (Note: The square root sign, √, refers only to the //positive// root. Use ± to include both roots.) | ||
Factorial: | Factorial: | ||
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==== Logs ==== | ==== Logs ==== | ||
- | inverse of a power | + | Log is the inverse of a power |
x = bᵉ, e = logᵦ(x) (b = base, 10 by default) | x = bᵉ, e = logᵦ(x) (b = base, 10 by default) | ||
eg. 2³ = 8, log₂(8) = 3 | eg. 2³ = 8, log₂(8) = 3 | ||
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==== Functions ==== | ==== Functions ==== | ||
+ | Definitions: | ||
+ | * Function: only has one y value for any x value. Discontinuities are okay (breaks in allowed x values) | ||
+ | * One-to-one function: a function with only one x value for any y value. | ||
+ | * Domain: what values of x are described | ||
+ | * Range: resulting values of y coming from the function | ||
composition of functions: (f’g)(x) = f(g(x)), order of evaluation is important | composition of functions: (f’g)(x) = f(g(x)), order of evaluation is important | ||
- | inverse of function - only possible if no two values of x produce the same result | + | |
- | graphing an inverse: reflection of the graph about the line y=x | + | **Inverse** of a function - only possible if no two values of x produce the same result, ie. must be one-to-one (or limit the domain to make it so) |
+ | Example: To find f⁻¹(x) for f(x) = 5x + 3 | ||
+ | y = 5x + 3 | ||
+ | y-3 = 5x | ||
+ | x = (y-3)/5 | ||
+ | f⁻¹(x) = (x-3)/5 (replace y with x on the last step, | ||
+ | since x is input to the function, and y is output) | ||
+ | |||
+ | Graphing an inverse: reflection of the graph about the line y=x | ||
**horizontal and vertical asymptotes:** | **horizontal and vertical asymptotes:** | ||
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====misc==== | ====misc==== | ||
abs. value of imaginary number | abs. value of imaginary number | ||
- | |a + bi| = sqrt(a^2 + b^2) | + | |a + bi| = √(a^2 + b^2) |
__rational numbers__ can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats. | __rational numbers__ can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats. |