Lesson 3-9: More On Functions

1. Lesson 3-9: More On Functions Objective Students will: Find composite functions Evaluate composite functions for a given value

2. Step Functions- a relationship that stays the same value for a set interval then steps up (or down) for the next interval. Example a cell phone’s minutes You may pay $0.15 per additional minute but what happens if you only talk 10 seconds? You pay for a full minute! What if you talk 5 minutes and 1 sec? You pay for 6 minutes! How would this look when graphed?

3. Let x = the number of Minutes Let y=Additional Charge 1.05 0.90 0.75 0.60 0.45 0.30 0.15 0 Cost 0 1 2 3 4 5 minutes

4. Questions: 1)Why is is referred to as a “step” function 2)For this example why are the negative quadrants not visible? 3) Why is the an open and closed circle at each integer?

5. Greatest Integer Function (another step function): notated y = [x] y = [x] means: the greatest integer that is less than or equal to x Ex: [4.6] = 4 [-1] = -1 [-2.8] = -3 Graph this function: Hint: think about where the function is opened or closed.

6. Remember: a negative input becomes positive Graphing Absolute Value Function: On the positive side it is just the line y=x The negative side also has a positive output An absolute value graph always takes on a V shape but… Where is the “bounce” point? (min or max)

7. Graph y = | x - 2 | What input will make the output zero? 2! Adding a number inside shifts the bounce point right. What would subtracting inside do??? Graph y = | x | + 3 What input will make the output zero? There isn’t one? Adding a number outside shifts the bounce point up. What would subtracting outside do???

8. Predict What would the graph of Look like?

9. Composite Functions • Combination of 2 or more functions like: f(x) and g(x) • Written: f(g(x)) → g(x) replaces x • Plug one function into x in the other • Since f is on outside g goes into the f function • f(x) = x + 2 g(x) = 3x • f(g(x)) = 3x + 2 • Evaluating Composites • Plugging in a number for the variable • 2 choices • Evaluate f(g(x)) first → then plug in the value (like above) f(g(5))= 3(5) +2 =17 2) Plug the number in g(x) first – evaluate; plug this number into f(x) – evaluate

10. 2) Plug the number in g(x) first – evaluate; plug this number into f(x) – evaluate f(g(5))= g(5) = 3(5) = 15 f(15) = 15+2 =17 You get the same answer either way!!!

11. You Try 1: f(x) = 2x – 1 , g(x) = 5x; find f(g(x)) You Try 2: g(x) = x2 – 1 , h(x) = x + 2; find g(h(x)); find g(h(3)) Hmmm… This one is challenging