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Chapter 1: Prerequisites for Calculus - Coggle Diagram

- - - - Equations of Lines(Connecting to the slope of a line): Equations of lines with a slope can be written in 3 ways--> Point-slope equation of lines, Slope-intercept equation of lines, General linear equation.
        Point-slope equation of lines: This is mostly used when you know the slope, m,(refer to 1.1(slope of a line), and the x and y coordinates of a point on the line. The equation is as follows: (y-y1)=m(x-x1), where y1 and x1 are the x and y coordinates of the point.
        Slope-intercept equation of lines: This form is used when you know the slope and the y-intercept(b) . The equation is as follows: y=mx+b
        you can also calculate slope when given 2 points on the line, using the formula for slope of a line
        There are 3 other questions of lines that can be used: For vertical lines, use x=a For horizontal lines, use y=b General Linear Equation: Ax+By=C or Ax+By-C=0
        
        Parallel & Perpendicular Equations: Parallel lines have the same slope(m), whereas perpendicular lines have slopes that are the inverse of each other.
        Eg. Parallel --> y=-3x+9, y=-3x+12
        Perpendicular--> y=-3x+2, y=1/3x+7
        
        Even and Odd functions: To determine whether a function is even or odd algebraically, use algebra to determine whether the following statements are true in the equation of the function given.
        For even: f(-x)=f(x), this means if you were to change all the x values in the function to (-x) and simplify, you would get back the original function.
        For odd: f(-x)= -f(x), this means if you were to change all the x values in the function to (-x) and simplify, you would get back the original function with a negative at the beginning.
        
        One-to-One function: to determine whether a function is one-to-one algebraically, use the following formula: fx1=fx2. If x1=-x2, it is one-to-one, otherwise it is not. Eg. f(x) =2x+5, 2x1+5=2x2+5(cancel out the 5 and then the 2, x1=x2, it is one-to-one
        
        Log rules: Refer to Exponential and Logarithmic functions for log rules to help solve Log functions algebraically
        
        Cartesian equations: Must be solved algebraically by cancelling out t somehow. Refer to Parametric Equations section.
- - - - Composition of Functions: When there are two functions and the range of one is included in the domain of the second, we can represent this using the composition of functions.
        Eg. (G o F)(x) = g(f(x), (G . F)(x) = g(x) x f(x)
        
        One-to-one functions: These functions cannot have the same y-coordinate for two different x coordinates. Use the horizontal line test to determine whether the graph of a function is a one-to-one. If the horizontal line runs through the Cartesian plane of the graph and only intersects once, then the function is -one-to-one.
      - Piecewise Function: These functions have a different formula for each part of its domain.
        for example, f(x)= x, for x>1
        =x-2, for x<1
        
        Absolute Value: When an absolute value is provided in the function, create the piecewise by using the given equation in the negative for x<1 and in the positive when x>1.
      - Interval Notation: Use interval notations as a way to represent domain and range. Closed intervals(where the value is included in the function), use [ and ], a closed circle
        Open intervals(where the value is not included(so only greater than or less than, not equal to), use ( and ) and an open circle. Open intervals are used when the values are not permitted in the function. This can be when there is a hole or asymptote in the function.