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Building Functions
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- 2. BUILDING FUNCTIONS CALIFORNIA CONTENTSTANDARDS Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 4. Find inverse functions. • Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x 1. • (+) Verify by composition that one function is the inverse of another. • (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
- 3. BUILDING FUNCTIONS RECALL! ▸What makesa function a function? ▸Turn to the partner next to you and take 1 minute to discuss the properties of a function!
- 4. BUILDING FUNCTIONS LET’S STARTOFF WITH AN EDUCATIONAL VIDEO https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=2ahUKEwi26dSgssPmAhX LsJ4KHTFpCCAQtwIwAXoECAcQAQ&url=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dw UNWjd4bMmw&usg=AOvVaw2ryROwHWXGkL35mqoVf-TT
- 5. BUILDING FUNCTIONS NOW TRYIT YOURSELF! ▸Find f(g(-3))! ▸f(x)=x^2-1 ▸g(x)=x+1
- 6. BUILDING FUNCTIONS HOW DOYOU COMPLETE THE PREVIOUS QUESTION? ▸Find f(g(-3)) ▸f(x)=x^2-1 ▸g(x)=x+1 ▸1) plug in -3 to g(x) ▸2) g(-3)=(-3)+1 ▸3) g(-3)=-2 ▸4) Then, plug this answer into f(x) ▸5) f(-2)=(-2)^2-1 ▸6) f(-2)=3 ▸7) f(g(-3))=3
- 7. BUILDING FUNCTIONS STEP BYSTEP HOW TO BUILD FUNCTIONS ▸f(g(x)) ▸ g(x) would be considered the “inside” function ▸ f(x) would be considered the “outside” function ▸Always work with one function at a time! ▸First plug in x to g(x) ▸Once you find your answer, plug in that answer to f(x) ▸It’s as easy as that!
- 8. BUILDING FUNCTIONS FINDING F(G(X))FROM A GRAPH ▸Find g(f(-2)) ▸f(x) = red graph ▸g(x) = blue graph ▸First, go to f(-2) and find what it equals (3) ▸Then, go to the graph of g(x) and find g(3) ▸g(3) = 4 so g(f(-2)) = 4
- 9. BUILDING FUNCTIONS NOW YOUTRY! ▸f(x) = red graph ▸g(x) = blue graph ▸Find f(g(-1))
- 10. BUILDING FUNCTIONS ANSWER ▸f(x) =red graph ▸g(x) = blue graph ▸Find f(g(-1)) ▸g(-1) = -1 ▸f(-1) = -2 ▸f(g(-1)) = -2
- 11. BUILDING QUESTIONS BONUS QUESTION ▸Nowfind a formula for f(x) (red graph) and g(x) (blue graph) ▸After finding a formula for each, find g(f(6)) ▸Turn to the person next to you if you need help
- 12. BUILDING FUNCTIONS ANSWER ▸f(x) =2x^2+3x-1 ▸g(x) = x^3 ▸Find g(f(6)) ▸f(6) = 2(6)^2+3(6)-1 ▸f(6) = 72+18-1 or 89 ▸g(89) = 89^3 or 704969 ▸g(f(6)) = 704969
- 13. BUILDING FUNCTIONS OTHER WAYSTO WRITE F(G(X)) ▸There is another way to write f(g(x)) ▸f(g(x)) = (f o g)(x) ▸Both of these would be pronounced “ f of g of x “
- 14. BUILDING FUNCTIONS ANOTHER WAYTO SOLVE ▸f(x) = x^2+1 ▸g(x) = x-1 ▸Solve for f(g(x)) ▸You can solve this without numbers by plugging the function g(x) into the output “x” for f(x) ▸This would look like f(x-1) or f(g(x)) = (x-1)^2+1 ▸This eliminates solving for g(x) first then f(x) and allows you to solve all at once
- 15. BUILDING FUNCTIONS YOUR LASTATTEMPT ▸Solve for f(g(2)) by plugging in g(x) to f(x) first. ▸f(x) = x^2-3x+4 ▸g(x) = x-6 ▸Solve by yourself and if you have any questions raise your hand!
- 16. BUILDING FUNCTIONS ANSWER ▸Solve forf(g(2)) by plugging in g(x) to f(x) first. ▸f(x) = x^2-3x+4 ▸g(x) = x-6 ▸f(g(x)) = (x-6)^2-3(x-6)+4 ▸f(g(2)) = (2-6)^2-3(2-6)+4 ▸f(g(2)) = 8
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