1. Not a function, function, one-to-one? How to draw an inverse given sketch Finding Inverses Domain and Range of f, f Domain of Composite Functions Graph Exponential Functions Verify it’s an inverse Evaluating Composite Functions Log Domains Graph Log Functions Solve Logs Interest Rates Exponential Growth Terms of a and b, Evaluate Logs Expand, Compress Logs Exponential Decay\ Half-life Please report any errors ASAP by email to sakim@fjuhsd.net or IM at kimtroymath. sakim@fjuhsd.net Problems may be more difficult on test. Consult homework assignment. Not all topics covered.

2. Type of relation. Draw inverse NOT A FUNCTION: fails vertical line test FUNCTION: pass vertical line test only. ONE-TO-ONE: passes vertical and horizontal line test. Be able to explain why it is a function, one-to- one, or not a function. Just like to find the inverse of coordinates we switch the x and y, we do the same with graphs. (, ) 34 2 (, ) -20 Remember, to draw freehand, it’s a reflection over the line y = x

3. Finding Inverses Switch the x and the y, solve for y. Important, when you have to find inverses of rational expressions that have fractions, you will probably have to cross multiply, move the y’s to one side and FACTOR! A couple of things to note, this was a case where f and f -1 happened to be the same, it doesn’t happen normally. The answers to the left are the same, it all just depends on which way you moved the terms.

4. Verify it’s an inverse. If it’s an inverse: (f ◦ f -1 )(x) = x and (f -1 ◦ f)(x) = x You will probably only be asked to verify in one direction though. (f ◦ f )(x) = x (f ◦ f)(x) = x Clear: (f ◦ f )(x) = x Clear: (f ◦ f)(x) = x Multiply by the common denominator to get rid of fractions! Careful when distributing. Be careful when distributing, be careful with signs. Multiply by the common denominator to get rid of fractions! Careful when distributing. Be careful when distributing, be careful with signs.

5. Domain and Range of f, f -1 Remember: Domain of f = range of f -1 Range of f = domain of f -1 Look at the previous three slides, I used the same problems. You can probably expect to have to find the inverse, verify, and find the domain and range given a function. Inverse Work

6. Evaluating Composite Functions f(x) g(x)

7. Domains of Composites 1) First, find domain of f(x) [note, letters changed, same process.] 2) Find domain of g(x), and find values of f(x) that equal the excluded values. 3) Combine in interval notation.

8. Graph Exponential Functions 1)Factor 2)Transformations 1)Compress\Stretch 2)Reflect 3)Shift 3)Key Points transformation 1) 4)Asymptote 1)y = 0 5)Graph Remember, reciprocal for horizontal compression stretch. a = 3 Asymptote follows vertical shift (up\down) Point Work Clear

9. Log Domains Everything inside the log is greater than zero. Check to see where it is zero AND undefined. Make a number line! -2 2 x = -3x = 0x = 3 ––+ Open circle, there is no ‘equals to’ Pick Positive Region, want greater than zero. REMEMBER: include the boundary points, use just use ( ), not [ ] -4 -2 2 x = -5 – x = -3 + x = 0 – x = 3 +

10. Graph Log Functions 1)Factor 2)Transformations 1)Compress\Stretch 2)Reflect 3)Shift 3)Key Points transformation 1) 4)Asymptote 1)x = 0 5)Graph Remember, reciprocal for horizontal compression stretch. a = _1_ 2 Asymptote follows horizontal shift (left\right) Point Work Clear Be warned, it is possible for both logarithmic and exponential you can have a fractional base. But you do the problem the same way. So don’t panic, we live in a beautiful world.

11. Terms of a and b, Evaluate logs. Terms of a and b problems Break down the log into factors Use the log rules to break them down Then substitute. Evaluating logs. Watch out for change of base formula. Sometimes, after you change the base, you might put it back together. These are the main types. Look at the tricky one on pg 308: 21, 22 also. The key to these problems is to know your rules and use order of operations properly. Remember, if there is an exponent, take care of the exponent first.

12. Solve Logs. Know your rules, don’t do anything illegal, and DOUBLE CHECK!!!! √√ Remember exact vs. approximate 2 logs, same base, try to make the insides equal. Logs with same base, number, use log rules to combine, then switch to exponential form. Remember to combine first. Exponential form, log both sides. You may have to FACTOR out x to help solve. This is an exact answer, if it’s on the calculator part, you may be asked to approximate. Also note, if the bases happen to be the same, you can just make the exponents equal to each other and solve. If you see something in the form below, you should probably solve by factoring. There are also other tricky examples such as Pg 313: 39 – 41, 45 – 49. It’s important that you follow log rules. Many of you broke the log rules on the red problem on the test. Watch out for that.

13. Expand (sum and difference of logs) Compress Logs (single log term) WATCH OUT FOR FACTORING, PARENTHESIS Watch out for factoring, such as the blue  green step. Be careful about how you split things apart.

14. Find the principal amount if the amount due after 4 years is $200 at 4% interest compounded continuously. Interest Rates Be able to: - Find amount due. - Principal amount. - Years invested. If it says ‘compounded continuously’ you should probably use the continuous compound interest formula. Suggestion: show set up incase you type it into your calculator incorrectly. What is the amount due after 2.25 years if $200 is invested at 3.2% compounded weekly? CAREFUL WITH CALCULATOR!!!!! Compound Interest P = 200  Principal r =.032  Interest Rate (decimal) n = 52  # of times compounded per year t = 2.25  Length of investment Continuous Compound Interest A = 200  Amount Due r =.04  Interest Rate (decimal) t = 4  Length of investment How long was the money invested for if the principal of $16000 returned $25000 at 4.75% interest compounded daily? Compound Interest P = 16000  Principal A = 25000  Amount Due r =.0475  Interest Rate (decimal) n = 365  # of times compounded per year. Also study effective interest rate. It may or may not show up. Be careful with rounding. Use approximate symbols where necessary.

15. Exponential Growth N 0 represents initial number of cells. k represents growth rate of cells. Amount at t = 0. Find growth rate. Find population after time t. Find how long it takes for something to reach a particular population. How long does it take to double, triple, multiply be some amount. Other various items. There are 4 bacteria in a culture that follows unbounded exponential growth. After 3 days, there are 10. What is the growth rate? This means find ‘k’ 10 = N(3)  Population after three days. 4 = N o  Initial Population 3 = t  Time (in days) Find the population after 5 days Round to the nearest whole number. Plug in 5 for t 4 = N o  Initial Population 5 = t  Time (in days).305 = k  Growth rate It is possible for me to ask the second question without the first. You would need to know that you need to figure out k first. (Different problem) How long until a population triples if the growth rate is 5% (units in days) 3N is because you triple the amount N. Don’t be thrown off by the wording. It could be bacteria, people, stress molecules, who knows. Understand what the questions are asking and what you need to find.

16. Exponential Decay N 0 represents initial number of cells. k represents decay rate of cells. Amount at t = 0. Find decay rate. Find population after time t. Find how long it takes for something to reach a particular population. How long does it take to cut in half, a third, some other amount. Other various items. Everything pretty much works the same as growth. The only different style problem involves half-life. I will show the long way to do the problem, and then the shortcut should be in the notes. The half life of radioactive isotope Kim-302 is 12 years. Find the decay rate of the Kim isotope.