Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Inverse Functions Name: Date:
Finding Inverse Functions – Guided Notes
We’ve explored the idea that the inverse of a function undoes the function. One way of
finding the inverse of a function is to _______________________________________________________. You
can think of this as switching the inputs and the outputs.
Example 1: Find the inverse of a(b) = 3b – 7
Example 2: Find the inverse of a(b) = 3b3 – 2b + 5
Example 3: Find the inverse of a(b) = 2x2 + 3x + 4
But the function in example 3 is not _________________________ so even though we can find an
inverse, that inverse is not a function. We can solve this problem by _____________________
____________________________ of both the function and its inverse.
Math 2 Week 4 Packet Page 1
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Inverse Functions Name: Date:
Similarly, we can also easily find the inverse of the graph of a function by switching the
inputs and the outputs. In graph form this looks like _____________________________________.
Example 4: Sketch the inverse of the graph shown below.
Example 5: Sketch the inverse of the graph shown below.
Is the inverse a function? Why or why not?
4
2
–2
–4
–5 5
4
2
–2
–4
–5 5
Math 2 Week 4 Packet Page 2
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Inverse Functions Name: Date:
Example 6: Sketch the inverse of the graph shown below.
Example 7: Sketch the inverse of the graph shown below.
Generalize: To sketch the graph of the inverse of a function, reflect the graph of the
function across the line _________________________________. This will switch the inputs and the
outputs.
Groupwork: In your groups, read and complete the work in the textbook beginning
with #2 on page 301 and ending with #4 on page 304.
Homework: Pages 306-‐208 # 3, 4 (a-‐f) (use your calculator), 5, and 7. The following
problems are reach problems: 3c, 3e, 4e, and 4f. All the rest are standard problems.
4
2
–2
–4
–5 5
4
2
–2
–4
–5 5
Math 2 Week 4 Packet Page 3
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Do Now 11.5 Name: Date:
Do Now 11.5 –Evaluating Functions
Consider the function f(x) = 3x3 – 2x2 + 5x – 4.
1. Find f(1)
2. Find f(2)
3. Find f(10)
4. Find f(f(3)
5. Find f-‐1(1444)
6. Find f-‐1(f-‐1(31082)
7. Find f(f(15)
8. Find f(20)
Math 2 Week 4 Packet Page 4
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Do Now 12 Name: Date:
Do Now 12 – Inverses of Functions
1. Find inverse functions for these functions.
a.
€
k(x) =4(x + 6)3
− 2 b.
€
v(x) =15 − x4
2. Sketch the graph of the inverse of each function on the same axes. Determine whether the inverse is a function or merely a relation. a.
b.
Is the inverse a function? Why or why not?
Is the inverse a function? Why or why not?
4
2
–2
–4
–5 5
4
2
–2
–4
–5 5
Math 2 Week 4 Packet Page 5
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Unit 1 Test Review Name: Date:
Unit 1 Test Review
Students will be able to (SWBAT):
• Write an explicit and a recursive function rule for a linear table of values.
• Describe in words or write a recursive function rule for a non-linear table of values.
• Determine whether a relation is a function based on a description, a table, or a graph.
• Find domain and range of a function from a graph.
• Based on an equation, determine domain restrictions for rational and radical functions.
• Find the inverse of a linear function.
• Given the graph of a linear function, graph its inverse.
• Evaluate compositions of functions, and find a rule for a composition.
1. Consider the functions f(x)=3x+7
g(x)=
€
x +12
Calculate each value
a. f(2) b. g(3) c. f(g(5)) d. g(f(2)) e.
€
f (2)⋅ g(3) f. f(2) + g(3)
Math 2 Week 4 Packet Page 6
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Unit 1 Test Review Name: Date:
2. Given the functions R(x) = x – 2 Q(x)=
€
x 2 + 4x +1 a. Find a formula for R(Q(x)) b. Find a formula for Q(R(x)) 3. State the domain of each function. You must write inequalities or use interval notation a.
€
b(x) = 2x 2 − 6 b.
€
c(x) = 2x − 6
Domain: Domain:
c.
€
d(x) =4
8 + x d.
€
e(x) =1+ x1+ x
Domain: Domain:
Math 2 Week 4 Packet Page 7
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Unit 1 Test Review Name: Date:
4. State the domain and range of each function. The functions do NOT extend beyond the graphs shown.. a.
Domain: Range:
b.
Domain: Range:
Math 2 Week 4 Packet Page 8
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Unit 1 Test Review Name: Date:
5. a. Complete the difference column for this table. b. Find a recursive function definition and an explicit (closed form) function definition that agree with the table below.
Input, n Output, p(n)
€
Δ
0 -8
1 -5
2 -2
3 1
4 4
Recursive definition:
€
p(n) =" # $
Explicit definition: p(n) =
Math 2 Week 4 Packet Page 9
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Unit 1 Test Review Name: Date:
6. For each function: a. Sketch a graph of the function below by making a table of values or using your graphing calculator. b. On the same graph, graph the inverse function or relation. Label both graphs to distinguish them. c. Is the inverse a function or merely a relation? d. Find the inverse function or relation.
f(x)=
€
x2
+ 3 g(x)=
€
8x − 4 h(x)=
€
0.5x 2 j(x)=
€
−3x
i. f(x)=
€
x2
+ 3
Graph f(x) and its inverse.
Is the inverse a function? Write the inverse using the correct name for it, in this case
€
f −1(x).
ii. g(x)=
€
8x − 4 Graph g(x) and its inverse.
Is the inverse a function? Write the inverse using the correct name for it.
Math 2 Week 4 Packet Page 10
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Unit 1 Test Review Name: Date:
iii. h(x)=
€
0.5x 2 Graph h(x) and its inverse.
Is the inverse a function? Write the inverse using the correct name for it.
iv. j(x)=
€
−3x Graph j(x) and its inverse.
Is the inverse a function? Write the inverse using the correct name for it.
Math 2 Week 4 Packet Page 11
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Unit 1 Test Review Name: Date:
7. a. Which of the following recursive functions best fits the table below?
Input, n Output,
a(n)
0 2
1 -1
2 -7
3 -19
4 -43
5
i.
€
a(n) =2 if n = 0
a(n −1) − 3 if n > 0# $ %
ii.
€
a(n) =2 if n = 0
2⋅ a(n −1) − 5 if n > 0$ % &
iii.
€
a(n) =2 if n = 0
a(n −1) + 2n − 5 if n > 0# $ %
iv.
€
a(n) =2 if n = 0
a(n −1) − 2n − 2 if n > 0# $ %
b. Fill in the value for a(5) in the table using the definition you chose. 8. Write an equation for a linear function with a slope of 4 and passing through the point (5, -2).
Math 2 Week 4 Packet Page 12
Math 2: Algebra 2, Geometry and Statistics
Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434
Name: Date:
Week 4 Assignment Guide
B-‐Block C-‐Block
Monday Do Now 11: Functions Pop Quiz In Class: Inverse Functions, Graphically and Algebraically. Homework: Pages 306-‐208 # 3, 4 (a-‐f) (use your calculator), 5, and 7. The following problems are reach problems: 3c, 3e, 4e, and 4f. All the rest are standard problems.
Do Now 11: Functions Pop Quiz
In Class: Inverse Functions, Graphically and Algebraically. Homework: Pages 306-‐208 # 3, 4 (a-‐f) (use your calculator), 5, and 7. The following problems are reach problems: 3c, 3e, 4e, and 4f. All the rest are standard problems.
Tuesday Do Now: DN 11.5 – Evaluating Functions In Class: Finding Zeros with Calculators, Evaluating Functions with Calculators, Begin Review if Time Homework: None
Thursday DN 12 – Inverses of Functions In Class: Test Review Homework: Finish Test Review
DN 12 – Inverses of Functions In Class: Test Review Homework: Finish Test Review
Friday In Class: Unit 1 Test Homework: None
In Class: Unit 1 Test Homework: None