 One of the more important tools is the ability to use function notation. And this emerges as follows. We frequently express the value of a quantity using this function notation. Some examples. We might talk about the velocity of a projectile in meters per second. Note that units are very important here. T seconds after an experiment begins. Or another possibility. We might look at the population of a country in millions of persons. T years after some designated starting year. Or possibly we may look at the percentage of voters who support a candidate. T days before an election. And in every case the function notation tells us something useful about the quantity. For example, let's let v of t be the velocity of an object in meters per second. T seconds after an experiment begins. And what is it telling us? Interpret v of zero equal to 25. So one very useful thing to do here is we can instantiate the definition by writing it down. And then replacing the terms of the definition with some specific values. So our definition v of t is the velocity of the objects in meters per second. T seconds after an experiment begins. Paper is cheap. Understanding is priceless. v of t equals. We'll write down that definition. Now what do we have? Well what we know is we have this v of zero. So t and zero. We're going to replace those two. And first step this is v of zero. T has been replaced by zero. This is the velocity of an object in meters per second. Zero seconds after an experiment begins. And then v of zero is the same thing as 25. So I'll make that next replacement. And so I have the 25 is the velocity of an object in meters per second. Zero seconds after the experiment begins. And so this suggests v of zero equals 25. We can read this as at the start of the experiment the velocity of the object is 25 meters per second. Well let's take a look at another example here. Let v of t equals 150 minus 5t. Here we have an explicit formula for the velocity of the object. We want to determine how rapidly the object is moving after six seconds. Well a little bit of analysis goes a long way. In this problem our function actually gives the velocity of an object in meters per second. And what we're looking for how rapidly is the object moving. Well that sounds a lot like a velocity. So we should write down our function expression. Note that we've included the units of that function expression. Again paper is cheap and it doesn't do any good to skimp on the writing when it makes the comprehension much more difficult. Now we've been given an amount in seconds and the only thing in the problem that's measured in seconds is going to be our t value. So that suggests that t is equal to 6. So we'll let t equal 6. We'll substitute that into our expression v of t v of 6 150 minus 5 times 6 meters per second. And after all the dust clears v of 6 equals 120 meters per second. And again we should state our answer including the actual units. The velocity after six seconds was 120 meters per second. Well here's another problem. The height of an object after t seconds is going to be given by some formula. When did the object hit the ground? Again a little bit of analysis goes a long way. In this problem we're given the height of an object. And so we know something about the height of the object. And we're looking for a when. So that suggests a time. And again what that suggests is we should begin by writing down our function expression. Once again notice that we've included the units of our expression. Now notice we've been given a condition when did the object hit the ground. And what that suggests is that we have the height of zero meters. So we'll substitute that in. Our height is zero meters. And we have, well actually this is a nice little quadratic equation. So we can now solve this equation. Our quadratic formula, simplification and our approximate values. And we finally go to our interpretation. t is the time after the object is fired from the cannon. So only the second solution makes sense. The first solution is a time before the object is fired. And that doesn't make a lot of sense. And so we can state our final answer including the units. The object hits the ground after 60.033 seconds. Well how about a graph? So I suppose f of x is the temperature of a location in degrees Fahrenheit x hours after 6 a.m. And now I have this graph showing the plot of y equals f of x. And we want to know when did the temperature drop below zero degrees Fahrenheit? Well f of x is the temperature. y equals f of x. So the question, when did the temperature drop below zero degrees? Well temperature, when did f of x, when did y drop below zero? And so there's the important connection we make. We want to look at the place where our y values dropped below zero. And if we look at our graph, that's going to occur right here at t equals 5. Not here because the temperature did not actually drop below zero there. Rather at t equals 5. And so at x equals 5, x is the number of hours after 6 a.m. So when did the temperature drop below 5 hours after 6 a.m. Or we could say 11 a.m.