This document discusses building new functions from existing functions. It provides examples of replacing f(x) with f(x) + k, kf(x), f(kx), and f(x + k) and examining the effects on the graph. Students are asked to find inverse functions and verify functions are inverses of one another through composition. Examples are provided of evaluating composite functions f(g(x)) from graphs and algebraic expressions by first applying the inner function g(x) and then the outer function f(x).

1. BUILDING
FUNCTIONS
MS. SCHREIBER
ALGEBRA II
9TH-12TH GRADE

2. BUILDING FUNCTIONS
CALIFORNIA CONTENT STANDARDS
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 makes a 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 START OFF 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 TRY IT YOURSELF!
▸Find f(g(-3))!
▸f(x)=x^2-1
▸g(x)=x+1

6. BUILDING FUNCTIONS
HOW DO YOU 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 BY STEP 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 YOU TRY!
▸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
▸Now find 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 WAYS TO 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 WAY TO 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 LAST ATTEMPT
▸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 for f(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

17. BUILDING FUNCTIONS
ANY QUESTION