Functions are machines. Plug the independent variable into the machine and it spits out the dependent variable.
Sample Problem
If y = f(x) = x + 1, then as x gets larger (moves right), y gets larger also (moves up). As x gets smaller (moves left), y gets smaller also (moves down).
Here's something to play with: see what happens to y as we make x larger or smaller.
Sample Problem
If y = f(x) = 1 – x, then as x gets larger (moves right), y gets smaller (moves down). As x gets smaller (moves right), y gets larger (moves up).
Sample Problem
Say y = f(x) = x2. We start x at -5. As x moves right, y gets smaller until x reaches 0. If, starting at 0, we keep moving x to the right, y starts getting bigger again.
With this function, in order to say whether y is increasing or decreasing as we play with x, we need to know two things: whether x is to the left or the right of zero, and whether x is being moved to the right or left.
Exercise 1
Let y = f(x) = -x2 + 1. If x is greater than zero, and getting larger, is y getting larger or smaller?
Exercise 2
Let y = f(x) = -x2 + 1. If x is greater than zero, and getting smaller, is y getting larger or smaller?
Exercise 3
Let y = f(x) = -x2 + 1. If x is less than zero, and getting smaller, is y getting larger or smaller?
Exercise 4
Let y = f(x) = -x2 + 1. If x is less than zero, and getting larger, is y getting larger or smaller?
Exercise 5
Let y = f(x) = -x2 + 1. As x gets close to zero, what does y approach?