inverse functions and relations · inverse functions and ... a2.2(d) use the composition of two...
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Inverse Functions and
Relations
LESSON 6–2
Over Lesson 6–1
5-Minute Check 6
A. f(1)
B. g(1)
C. (g ○ f)(1)
D. f(0)
Let f(x) = x – 3 and g(x) = x2. Which of the following is equivalent to (f ○ g)(1)?
TEKS
Targeted TEKS
A2.2(D) Use the composition of two
functions, including the necessary
restrictions on the domain, to determine if the
functions are inverses of each other.
Also addresses A2.2(C).
Mathematical Processes
A2.1(E), A2.1(G)
Then/Now
• Find the inverse of a function or relation.
• Determine whether two functions or relations are inverses.
Vocabulary
• inverse relation
• inverse function
Example 1
Find an Inverse Relation
GEOMETRY The ordered pairs of the relation {(1, 3), (6, 3), (6, 0), (1, 0)} are the coordinates of the vertices of a rectangle. Find the inverse of this relation. Describe the graph of the inverse.
To find the inverse of this relation, reverse the coordinates of the ordered pairs. The inverse of the relation is {(3, 1), (3, 6), (0, 6), (0, 1)}.
Example 1
Find an Inverse Relation
Answer: Plotting the points shows that the ordered pairs also describe the vertices of a rectangle. Notice that the graph of the relation and the inverse are reflections over the graph of y = x.
Example 1
A. cannot be determined
B. {(–3, 4), (–1, 5), (2, 3), (1, –2)}
C. {(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)}
D. {(4, –3), (5, –1), (3, 2), (1, 1), (1, –2)}
GEOMETRY The ordered pairs of the relation {(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)} are the coordinates of the vertices of a pentagon. What is the inverse of this relation?
Concept
Example 2
Find and Graph an Inverse
Step 1 Replace f(x) with y in the original equation.
Then graph the
function and its inverse.
Step 2 Interchange x and y.
Example 2
Find and Graph an Inverse
Step 3 Solve for y.
Inverse
Step 4 Replace y with f –1(x).
y = –2x + 2 f –1(x) = –2x + 2
Multiply each side by –2.
Add 2 to each side.
Example 2
Find and Graph an Inverse
Example 2
Find and Graph an Inverse
Answer:
Example 2
A.
B.
C.
D.
Graph the function
and its inverse.
Concept
Example 2
Step 1 Replace f(x) with y in the original equation.
Step 2 Interchange x and y.
Find the inverse of f(x) = x2 – 4x + 1. Then
graph the function and its inverse. If
necessary, restrict the domain of the inverse
to that it is a function.
Inverses with Restricted Domains
Example 2
Find and Graph an Inverse
Step 3 Solve for y.
Inverse
Subtract 1 from each side.
Complete the square
Simplify
Take the square root of each side.
Add 2 to each side.
Example 2
Find and Graph an Inverse
If the domain is restricted to [4, ∞), find the inverse of
f(x) = x2 – 8x + 10.
Check to see if the compositions of f(x) and g(x) are identity functions.
Verify that Two Functions are Inverses
Answer: The functions are inverses since both [f ○ g](x) and [g ○ f](x) equal x.
A. They are not inverses since [f ○ g](x) = x + 1.
B. They are not inverses since both compositions equal x.
C. They are inverses since both compositions equal x.
D. They are inverses since both compositions equal x + 1.
Inverse Functions and
Relations
LESSON 6–2