Functions

Things one must know:

• A function rule describes a series of operations to perform on a variable.
• The "domain" of a function indicates the possible inputs, or the x-coordinate. The "range" of a function indicates the possible "outputs", or the y-coordinate.
• Input the value in place of the variable to determine the value of the function. When substituting a variable expression into function, keep the expression inside parentheses.
• The key to solving compound function problems is to work from the inside out - i.e. use the result from the inner function as the new input variable for the outer function. 
• Changing the order of the compound functions changes the answer, to find a value of x for which f(g(x)) = g(f(x)), use variable substitution.
• Finding an unknown constant:
  1) Use the value of the input variable and the corresponding output value of the function to solve for the unknown constant.
  2) Rewrite the function, replacing the constant with its numerical value.
  3) Solve the function for the new input variable.
• Direct proportionality means that the two quantities always change by the same factor and in the same direction: y = kx, where x is the input value, y is the output value and k is the proportionality constant. Can also be written as y / x = k. Set up ratios for the before case and the after case, and then set the ratios equal to each other to solve.
• Inverse proportionality means that the two quantities change by reciprocal factors: y = k / x, where x is the input value, y is the output value and k is the proportionality constant. Can also be written as xy = k. This time set up products to solve the problem. 
• Linear growth - growth at a constant rate, same constant added each period: y = mx + b, where slope m is the constant rate at which the quantity grows, the y-intercept b is the value of the quantity at time zero and the variable x stands for time. Initial time is 0.
• Exponential growth - a quantity is multiplied by the same constant each period of time: y(t) = y₀·k^t, where y₀ represents value at time zero, k represents the constant multiplier (commonly in percentage form), and t represents time. Maybe it is y = k(R^t)
  
• Symmetry, the property that two seemingly different inputs to the function always yield the same output. Solve algebraically or it might be easier to pick a number.
• Pick numbers for advanced function problems that test whether certain functions follow certain properties of mathematics.
• Optimization
  1) Linear functions have straight line graphs. The extremes (max and min) of linear functions occur at the boundaries: at the smallest possible x and at the largest possible x, as given in the problem.
  2) Quadratic functions form parabolas. A parabola has either a peak (a max value) or a valley (a min value) that you need to find. The key is to make the squared expression equal to 0. Whatever value for x makes the squared expression equal to zero is the value of x that minimizes or maximizes the function. The sign of the squared term determines whether the extreme point is a minimum (positive sign) or a maximum (negative sign).