2. Inverses are everywhere – when we think about reversing a process, we are thinking about the inverse.

3. What does this mean mathematically? The BIG IDEA with inverse functions is that you switch the relationship around – essentially SWITCH THE X & Y.

4. This idea can be used to find the inverse equation of a function. Step1: rewrite the f(x) as y Step2: Interchange x and y. Step 3: Solve for y. This can involve some tricky algebra – see the next few slides for the steps on this example.

5. Solving for y Multiply both sides by (y – 3) and distribute the x to both terms. Get the “y” terms on one side by themselves (in other words move the 3x to the right and the 2y to the left).

6. almost there… Now to really get “y” alone you have to factor it out. Finally you divide both sides by (x – 2) And rewrite your equation with inverse function notation.

7. And wait – we aren’t done – let’s examine domain & range.. What did we say the BIG IDEA was with inverse? YUP – switching the x & y!

8. So let’s examine the domain & range in that previous example... Original function Domain Range Inverse function Domain Range As expected the domain & range switch in the inverse.

9. IMPORTANT FACT ABOUT INVERSES The domain of the inverse equals the range of the original! But don’t assume that the range of the inverse equals the domain of the original. Sometimes it doesn’t…..

10. When are inverses of functions also functions? Let’s look at f(x) = x 2 …. What is the inverse o this function? Technically if we reflect it over the line y = x, it is NOT a function.

11. Check it out in your calculator Run [DEFAULTS] [Y=] in Y1 type x 2 then do [GRAPH] Now do [2 nd ][draw] #8 DrawInv (don’t press enter yet – you’ll see DrawInv on your homescreen. If you press enter you get an error) Now type [VARS] ->Y-VARS [1] [1] [ENTER] Cool – you should see the inverse & the original function together on one screen! Hopefully you clearly see that the inverse is not a function.

12. The function f(x) = x 2 …. Does not pass the “Horizontal Line Test”

13. Horizontal Line Test? If any horizontal line passes through a function graph in only one point, that function is one-to- one. One-to-one functions have inverses that are also functions.

14. Horizontal Line Test? If any horizontal line passes through a function graph in MORE than one point, that function is NOT one- to-one

15. Interesting Fact about Horizontal Line Test For a function to be one-to-one (passes the horizontal line test), it must be everywhere INCREASING or everywhere DECREASING. These type of functions are also called MONOTONIC (a terms used in Calculus)

16. Horizontal Line Test for Inverses A function will have an inverse that is ALSO a function IFF the original function is one-to-one (passes the horizontal line test)

17. Okay officially the inverse of f(x) = x 2 is not a function – but it’s so important that mathematicians made the inverse exist by RESTRICTING THE DOMAIN. This happens with a few different functions (we will revisit this with sine, cosine & tangent).

18. Original function Domain Range Inverse function Domain Range Notice that important fact from before – THE DOMAIN OF THE INVERSE EQUALS THE RANGE OF THE ORIGINAL.

19. This is sucn an important statement: The DOMAIN OF THE INVERSE WILL ALWAYS BE THE RANGE OF THE ORIGINAL. But as usual with range of the inverse, you can’t make assumptions – always examine the graph to verify, it may not always be the same as the domain of the original (as seen in the quadratic inverse previously).

20. Okay – here’s another weird thing…. Given the original function: What is the inverse? Also state the domain & range…

21. original function: Inverse function: Careful – the new equation may lead you to think you’ve got a parabola for the inverse – but as the inverse of a square root function – remember the DOMAIN OF THE INVERSE IS THE RANGE OF THE ORIGINAL.

22. original function: Inverse function: REMEMBER graphically the inverse is a reflection over the line y = x.

23. Original function Domain Range Inverse function Domain Range Yup – we got the full switch of the domain & range for the inverse.

24. The Inverse Composition Rule You can test whether two function equations represent inverses by testing them. f ( g (x) ) = x g ( f (x) ) = x Cool – so this means you can check if two equations are inverses by doing the composition on BOTH functions (in both directions to be doubly sure!)

25. To verify if you have inverses…. Algebraically you must show BOTH f ( g (x) ) = x and g ( f ( x) ) = x Applying this rule is considered mathematical proof of the two function equations being inverses.

26. Verify that f(x) & g(x) are inverses using the COMPOSITION RULE (both directions!)

27. f(g(x) =

28. g(f(x) =

29. Let’s visit the BIG IDEA with a relation and its inverse The relation 9x 2 + 4y 2 – 18x + 18y = 23 is an ELLIPSE. Is it a function? But we can still talk about its inverse… If (1, -4) is a point on the ellipse, what is a point on the inverse of this ellipse? ABOSOLUTELY NOT!

30. First instinct I know is to somehow come up with the function equation for the inverse… 9x 2 + 4y 2 – 18x + 18y = 23 DON’T DO IT! Instead remember that simple fact about inverses – the x & y switch. Just switch around the ordered pair (1, -4) to be (-4, 1). And test it in the original equation. If it is a point in the inverse the “switched” ordered pair should work in the original.

31. Some classwork problems for you to try & check (answers at end) FOR EACH: find f -1 (x) and its domain & range Ex 1 f(x) = Ex 2 f(x) = x 2 + 4

32. answers… 1) f(x) = inverse f -1 (x) = 2) f(x) = x 2 + 4 f -1 (x) = Domain (-∞, ½) υ (½, ∞) Range (-∞, ¾ ) υ (¾, ∞) Domain [4, ∞) Range [0, ∞)

33. homework pp 135 – 137 #s 13 – 21 odd, 31, 35, 36, 41, 42, 47