Worked example: evaluating piecewise functions

consider the following piecewise function and they say f of t is equal to and they tell us what it's equal to based on what T is so if T is less than or equal to negative 10 we use this case if T is between negative 10 and negative 2 we use this case and if T is greater than or equal to negative 2 we use this case and then they ask us what is the value of f of negative 10 so T is going to be equal to negative 10 so which case do we use so let's see if T is less than or equal to negative 10 we use this top case right over here and T is equal to negative 10 that's the one that we're trying to evaluate so we want to use this case right over here so f of negative 10 is going to be equal to negative 10 everywhere where we see a T here we substitute it with a negative 10 negative 10 squared minus 5 times so actually I don't have a denominator there I don't know I wrote it so high so it's going to be negative 10 squared minus 5 times negative 10 so let's see negative 10 squared that's positive 100 and the negative or subtracting 5 5 times negative 10 this is going to be subtracting negative 50 or you're going to add 50 so this is going to be equal to 150 F of negative 10 is 150 because we use this case up here because T is negative 10 let's do another one of these examples so here we have consider the following piecewise function alright what is the value of H of negative 3 see when H is negative 3 which case do we use we'll use this case if if our X is between negative infinity and zero and negative 3 is it between is between negative infinity and zero so we're going to use this case right over here if it was positive 3 we would use this case if it was positive 30 we would use this case so we're going to use the first case again and so we're going so for H equal 4 H of negative 3 we're going to take negative 3 to the third power so let's see H of negative 3 is going to be negative 3 to the third power which is negative 27 and we're done that's H of negative three because we are using this case you could almost just ignore the second two cases right over here let's do one more example this one's a little bit different below is a graph of the step function G of X so we could see G of X right over here it started when x equals negative 9 it's at 3 and it jumps up and then jumps down match each expression with its value so G of negative three point zero zero zero one so negative three point zero zero zero one so that's right over here and G of that we see is equal to three so this is going to be equal to three right over here G of three point nine nine nine nine nine three point nine nine nine nine nine almost almost four so let's draw the dotted line right over here is going to be almost four well G of three point nine nine nine nine nine is going to be seven we see that right over there so that is equal to seven G of four point zero zero zero zero one so G of four is still seven but as soon as we go above four we drop down over here so G of four point zero zero zero zero one is going to be negative three I want to actually let's let's focus on that a little bit more how did I know that well I know that G of four is seven and not negative three because we have this dot is circled in up here and it's hollow down here but as soon as we get any amount larger than four then the function dropped down to this so four point zero zero zero zero as many you know just slightly above for the value of our function is going to be negative three now let's do G of nine so G of nine so when X is 9 we go down here you might be tempted to say it's negative three but you see at this point right over here we have an open circle so that means that well it's not it you can't say that the function is negative three right over there and there's no other place where we have a filled in circle for x equals nine so the function G actually isn't defined at x equals nine so I'm going to put undefined right over there