Consider the function g(x) shown below over a domain of [1,∞).
g(x) = 2(x – 1)2 – 3
Part A
Determine g–1(x).
Part B
Identify the domain and range of g(x) and g–1(x). Represent the domains and ranges in inequality, interval, and set notation. Describe the relationship between the domains and ranges of g(x) and g–1(x).
Part C
Graph g(x) and g–1(x). Describe the relationship between the graphs of g(x) and g–1(x).
Answers
The inverse of a function may or may not be a function
The function g(x) is given as:
[tex]g(x) = 2(x - 1)^2 - 3[/tex]
(a) Inverse function g^-1(x)
We have:
[tex]g(x) = 2(x - 1)^2 - 3[/tex]
Replace g(x) with y
[tex]y = 2(x - 1)^2 - 3[/tex]
Swap the positions of x and y
[tex]x = 2(y - 1)^2 - 3[/tex]
Add 3 to both sides
[tex]x + 3 = 2(y - 1)^2[/tex]
Divide both sides by 2
[tex]\frac{x + 3}2 = (y - 1)^2[/tex]
Take square roots of both sides
[tex]\sqrt{\frac{x + 3}2} = y - 1[/tex]
Add 1 to both sides
[tex]1 + \sqrt{\frac{x + 3}2} = y[/tex]
Rewrite as:
[tex]y = 1 + \sqrt{\frac{x + 3}2}[/tex]
Express as inverse function of g
[tex]g^{-1}(x) = 1 + \sqrt{\frac{x + 3}2}[/tex]
Hence, the inverse function of g is [tex]g^{-1}(x) = 1 + \sqrt{\frac{x + 3}2}[/tex]
(b) The domain, and the range of g(x) and g^-1(x)
Function g(x)
The function g(x) is a quadratic function.
So, the domain is:
Inequality: [tex]-\infty < x < \infty[/tex]Interval: [tex](-\infty ,\infty)[/tex]Set: {R}The range is:
Inequality: [tex]f(x) \ge -3[/tex]Interval: [tex][-3 ,\infty)[/tex]Set: [tex]\{y|y\ge -3\}[/tex]Function g^-1(x)
The function g^-1(x) is a square root function.
So, the domain is:
Inequality: [tex]x \ge -3[/tex]Interval: [tex][-3 ,\infty)[/tex]Set: [tex]\{x|x\ge -3\}[/tex]The range is:
Inequality: [tex]-\infty < y < \infty[/tex]Interval: [tex](-\infty ,\infty)[/tex]Set: {R}The domain of g(x) is the range of g^-1(x), while the range of g(x) is the domain of g^-1(x).
(c) The graph
See attachment for the graph
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Step-by-step explanation:
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[tex]\huge \bf༆ Answer ༄[/tex]
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Step-by-step explanation:
I celebrate on christmas eve :)
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Step-by-step explanation:
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Answer:
x 2-2 3-3
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Step-by-step explanation:
Each domain, or x value, can only be matched with one y value, so given this, the third one is the answer since every domain is only matched with one range, or in other words, there are no repeating domains in on the x row of the table.
The table that represents a function is
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Option D is the correct answer.
What is a function?A function has an input and an output.
A function can be one-to-one or onto one.
It simply indicated the relationships between the input and the output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
A function can have many to one but a function can not have one to many.
So,
From the table,
x 2 -2 4 -4
y 8 8 12 12
This is a many-to-one function.
All the other tables are one-to-many functions.
Thus,
The table that represents a function is
x 2 -2 4 -4
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Rhombus
Step-by-step explanation:
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Answer:
1st one is 68 and the 2nd one is 25. hope that helps
Answer:
1. 112°2. 25°Step-by-step explanation:
1. 50° + 62° = x°
112° = x°
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x° = 150° - 125°
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Answer:
Yes.
Step-by-step explanation:
Answer:
no.
Step-by-step explanation:
-4/19 is completely different to 4/-19.
Im not sure how to word it but they are two completely different numbers.
he first step that Sarah performed to simplify the expression StartFraction 4(6.8 minus 4.2) plus 8.2 over 2 EndFraction before entering it into the calculator is shown.
2(3.4 – 2.1) + 4.1
Which statement is true?
Sarah performed the first step correctly.
The first step should have been 2(6.8 – 4.2) + 8.2.
The first step should have been StartFraction 27.2 minus 4.2 plus 8.2 over 2 EndFraction.
The first step should have been StartFraction 4(2.6) plus 8.2 over 2 EndFraction.
Answers
Sarah performed the first step correctly.
Given the function that Sarah performed as:
[tex]\dfrac{4(6.8-4.2)+8.2}{2}[/tex]
To simplify the function, the first step Sarah carries out is to divide the numerator by 2:
[tex]\dfrac{4(6.8-4.2)+8.2}{2} \\= \frac{2(2)(2(3.4)-2(2.1)+2(4.1))}{2}\\=2(3.4 - 2.1) + 4.1[/tex]
This shows that Sarah performed the first step correctly.
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Answer:
i think its 4*1
Step-by-step explanation:
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Explainations