holt algebra 2 9-5 functions and their inverses determine whether the inverse of a function is a...
TRANSCRIPT
![Page 1: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/1.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Determine whether the inverse of a function is a function.
Write rules for the inverses of functions.
Objectives
![Page 2: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/2.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
one-to-one function
Vocabulary
When both a relation and its inverses are functions, the relation is called a one-to-one function.
In a one-to-one function, each y-value is paired with exactly one x-value.
![Page 3: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/3.jpg)
Holt Algebra 2
9-5 Functions and Their InversesRecall that the vertical-line test can help you determine whether a relation is a function.
Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one point.
Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.
![Page 4: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/4.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
04/21/23 03:24
1-6 Relations and Functions
4
Vertical Line Test
Would this graph be a function?YES
![Page 5: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/5.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
04/21/23 03:24
1-6 Relations and Functions
5
Vertical Line Test
Would this graph be a function?
NO
![Page 6: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/6.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Use the horizontal line test to decide if the inverse is a function.
![Page 7: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/7.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Use the horizontal-line test to determine whether the inverse of the blue relation is a function.
Example 1A: Using the Horizontal-Line Test
The inverse is a function because no horizontal line passes through two points on the graph.
![Page 8: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/8.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Use the horizontal-line test to determine whether the inverse of the red relation is a function.
Example 1B: Using the Horizontal-Line Test
The inverse is a not a function because a horizontal line passes through more than one point on the graph.
![Page 9: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/9.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Use the horizontal-line test to determine whether the inverse of each relation is a function.
The inverse is a function because no horizontal line passes through two points on the graph.
Check It Out! Example 1
![Page 10: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/10.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Recall from Lesson 7-2 that to write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.
![Page 11: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/11.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Example 2: Writing Rules for inverses
Find the inverse of . Determine whether it is a function, and state its domain and range.
Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.
![Page 12: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/12.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Example 2 Continued
Rewrite the function using y instead of f(x).
Step 1 Find the inverse.
Simplify.
Switch x and y in the equation.
Cube both sides.
Isolate y.
![Page 13: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/13.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Example 2 Continued
Because the inverse is a function, .
The domain of the inverse is the range of f(x):{x|x R}.
The range is the domain of f(x):{y|y R}.
Check Graph both relations to see that they are symmetric about y = x.
![Page 14: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/14.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Find the inverse of f(x) = x3 – 2. Determine whether it is a function, and state its domain and range.
Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.
Check It Out! Example 2
![Page 15: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/15.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Rewrite the function using y instead of f(x).
Step 1 Find the inverse.
Take the cube root of both sides.
Switch x and y in the equation.
Add 2 to both sides of the equation.
Simplify.
Check It Out! Example 2 Continued
y = x3 – 2
x = y3 – 2
x + 2 = y3
3 x + 2 = y
33 3x + 2 = y
![Page 16: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/16.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
The domain of the inverse is the range of f(x): R.
The range is the domain of f(x): R.
Check Graph both relations to see that they are symmetric about y = x.
Check It Out! Example 2 Continued
Because the inverse is a function, .
![Page 17: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/17.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value.
You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compose two inverses the result is the input value x.
![Page 18: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/18.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
![Page 19: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/19.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Determine by composition whether each pair of functions are inverses.
Example 3: Determining Whether Functions Are Inverses
Find the composition f(g(x)).
f(g(x)) = 3( x + 1) – 1 1 3
Use the Distributive Property.
Simplify.
f(x) = 3x – 1 and g(x) = x + 1 1 3
Substitute x + 1 for x in f.
1 3
= (x + 3) – 1
= x + 2
![Page 20: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/20.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)).
Example 3 Continued
Check The graphs are not symmetric about the line y = x.
![Page 21: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/21.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Example 3B: Determining Whether Functions Are Inverses
Find the compositions f(g(x)) and g(f (x)).
For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1 x
1x – 1
Because f(g(x)) = g(f (x)) = x for all x but 0 and 1, f and g are inverses.
= x
= (x – 1) + 1
= x
![Page 22: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/22.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Example 3B Continued
Check The graphs are symmetric about the line y = x for all x but 0 and 1.
![Page 23: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/23.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Check It Out! Example 3a
Determine by composition whether each pair of functions are inverses.
Find the composition f(g(x)) and g(f(x)).
3 2
f(x) = x + 6 and g(x) = x – 9 2 3
= x – 6 + 6
= x
f(g(x)) = ( x – 9) + 6 3 2
2 3
g(f(x)) = ( x + 6) – 9 2 3
3 2
= x + 9 – 9
= x
Because f(g(x)) = g(f(x)) = x, they are inverses.
![Page 24: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/24.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Check The graphs are symmetric about the line y = x for all x.
Check It Out! Example 3a Continued
![Page 25: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/25.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Find the compositions f(g(x)) and g(f(x)).
Simplify.
Check It Out! Example 3b
Substitute for x in f.
f(g(x)) = + 5
f(x) = x2 + 5 and for x ≥ 0
= x + 25 +5 10 x
= x – 10 x + 30
![Page 26: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/26.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)).
Check The graphs are not symmetric about the line y = x.
Check It Out! Example 3b Continued
![Page 27: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/27.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Lesson Quiz: Part I
A: yes; B: no
1. Use the horizontal-line test to determine whether the inverse of each relation is a function.
![Page 28: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/28.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
Lesson Quiz: Part II
D: {x|x ≥ 4}; R: {all Real Numbers}
2. Find the inverse f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.
not a function
![Page 29: Holt Algebra 2 9-5 Functions and Their Inverses Determine whether the inverse of a function is a function. Write rules for the inverses of functions. Objectives](https://reader035.vdocuments.us/reader035/viewer/2022081503/56649e4e5503460f94b44d10/html5/thumbnails/29.jpg)
Holt Algebra 2
9-5 Functions and Their Inverses
3. Determine by composition whether f(x) =
3(x – 1)2 and g(x) = +1 are inverses
for x ≥ 0.
Lesson Quiz: Part III
yes