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Apply the Horizontal Line Test
A. Graph the function f (x) = 4x 2 + 4x + 1 using a
graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.
The graph of f (x) = 4x 2 + 4x + 1
shows that it is possible to find a horizontal line that intersects the graph of f (x) more than once. Therefore, you can conclude that f –1 does not exist.
Answer: no
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Apply the Horizontal Line Test
B. Graph the function f (x) = x 5 + x
3 – 1 using a graphing calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.
The graph of f (x) = x 5 + x
3 – 1 shows that it is not possible to find a horizontal line that intersects the graph of f (x) more than one point. Therefore, you can conclude that f –1 exists.
Answer: yes
![Page 4: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/4.jpg)
Graph the function using a graphing
calculator, and apply the horizontal line test to determine whether its inverse function exists. Write yes or no.
A. yes
B. yes
C. no
D. no
![Page 5: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/5.jpg)
![Page 6: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/6.jpg)
Find Inverse Functions Algebraically
A. Determine whether f has an inverse function for
. If it does, find the inverse function
and state any restrictions on its domain.
The graph of f passes the horizontal line test. Therefore,
f is a one-one function and has an inverse function.
From the graph, you can see that f has domain
and range .
Now find f –1.
![Page 7: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/7.jpg)
Find Inverse Functions Algebraically
![Page 8: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/8.jpg)
Find Inverse Functions Algebraically
Original function
Replace f(x) with y.
Interchange x and y.
2xy – x = y Multiply each side by 2y – 1. Then apply the Distributive Property.
2xy – y = x Isolate the y-terms.
y(2x –1) = x Factor.
![Page 9: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/9.jpg)
Find Inverse Functions Algebraically
Divide.
![Page 10: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/10.jpg)
Find Inverse Functions Algebraically
Answer: f –1 exists;
From the graph, you can see that f –1 has domain
and range . The
domain and range of f is equal to the range and
domain of f –1, respectively. Therefore, it is not
necessary to restrict the domain of f –1.
![Page 11: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/11.jpg)
Find Inverse Functions Algebraically
B. Determine whether f has an inverse function for . If it does, find the inverse function and state any restrictions on its domain.
The graph of f passes the horizontal line test. Therefore, f is a one-one function and has an inverse function. From the graph, you can see that f has domain and range . Now find f
–1.
![Page 12: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/12.jpg)
Find Inverse Functions Algebraically
Original function
Replace f(x) with y.
Interchange x and y.
Divide each side by 2.
Square each side.
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Find Inverse Functions Algebraically
Add 1 to each side.
Replace y with f –1(x).
From the graph, you can see that f –1 has domain
and range . By restricting the domain of f –1 to ,
the range remains . Only then are the domain and
range of f equal to the range and domain of f –1,
respectively. So, .
![Page 14: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/14.jpg)
Find Inverse Functions Algebraically
Answer: f –1 exists with domain ;
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Determine whether f has an inverse function for
. If it does, find the inverse function and
state any restrictions on its domain.
A.
B.
C.
D. f –1(x) does not exist.
![Page 16: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/16.jpg)
![Page 17: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/17.jpg)
Verify Inverse Functions
Show that f [g (x)] = x and g [f (x)] = x.
![Page 18: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/18.jpg)
Verify Inverse Functions
Because f [g (x)] = x and g [f (x)] = x, f (x) and g (x) are inverse functions. This is supported graphically because f (x) and g (x) appear to be reflections of each other in the line y = x.
![Page 19: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/19.jpg)
Verify Inverse Functions
Answer:
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Show that f (x) = x 2 – 2, x 0 and
are inverses of each other.
A. B.
C. D.
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Find Inverse Functions Graphically
Use the graph of relation A to sketch the graph of its inverse.
![Page 22: Key Concept 1. Example 1 Apply the Horizontal Line Test A. Graph the function f (x) = 4x 2 + 4x + 1 using a graphing calculator, and apply the horizontal](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bfbd1a28abf838ca2214/html5/thumbnails/22.jpg)
Answer:
Find Inverse Functions Graphically
Graph the line y = x. Locate a few points on the graph of f (x). Reflect these points in y = x. Then connect them with a smooth curve that mirrors the curvature of f (x) in line y = x.
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Use the graph of the function to graph its inverse function.
A.
B.
C.
D.