Given any two functions f and g, which of the following statements is always true?(1 point)
Responses
If x is in the domain of f, then x must be in the domain of f−g.
If x is in the domain of , f, , then x must be in the domain of , f minus g, .
If x is not in the domain of fg, then x is not in the domain of g.
If x is not in the domain of , StartFraction f over g EndFraction, , then x is not in the domain of , g, .
If x1 is in the domain of f and x2 is in the domain of g, then the product, x1x2, must be in the domain of fg.
If , x subscript 1 baseline, is in the domain of , f, and , , x subscript 2 baseline, is in the domain of , g, , , then the product,, , x subscript 1 baseline x subscript 2 baseline, , must be in the domain of , f g, .
If x is in the domain of f+g, then x must be in the domain of f and x must be in the domain of g.
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What is (f−g)(x)?(1 point)
Responses
It is the difference of two functions, f(x)−g(x) times x.
It is the difference of two functions, , f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis, times x.
It is the product of two functions, f(x)⋅g(x).
It is the product of two functions, , f left parenthesis x right parenthesis times g left parenthesis x right parenthesis, .
It is the difference of two functions, f(x)−g(x).
It is the difference of two functions, , f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis, .
It is the difference of two functions, g(x)−f(x).
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Find fg and explain the domain for the function.
f(x)=2x2−25x+72, g(x)=x−8
(1 point)
Responses
The function is (fg)(x)=2x+9. The domain of fg is all real numbers because the domain of f and the domain of g are both all real numbers.
, The function is , left parenthesis StartFraction f over g EndFraction right parenthesis left parenthesis x right parenthesis equals 2 x plus 9, . The domain of , StartFraction f over g EndFraction, is all real numbers , , because the domain of f and the domain of g are both all real numbers.
The function is (fg)(x)=2x−9. The domain of fg is all real numbers except for x=8 because the denominator is 0 when x=8.
The function is , left parenthesis StartFraction f over g EndFraction right parenthesis left parenthesis x right parenthesis equals 2 x minus 9, . The domain of , StartFraction f over g EndFraction, is all real numbers except for , x equals 8, because the denominator is 0 when , x equals 8, .
The function is (fg)(x)=2x+9. The domain of fg is all real numbers except for x=8 and x=92 because the denominator is 0 when x=8 or when x=92.
, The function is , left parenthesis StartFraction f over g EndFraction right parenthesis left parenthesis x right parenthesis equals 2 x plus 9, . The domain of , StartFraction f over g EndFraction, is all real numbers except for , x equals 8, and , x equals 9 halves, because the denominator is 0 when , x equals 8, or when , x equals 9 halves, , .
The function is (fg)(x)=2x−9. The domain of fg is all real numbers because the domain of f and the domain of g are both all real numbers.
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Let f(x)=4x+5−−−−−√ and g(x)=25x−9−−−−−−√. Evaluate the expression.
(f−g)(1)
(1 point)
Responses
(f−g)(1)=1
left parenthesis f minus g right parenthesis left parenthesis 1 right parenthesis equals 1
(f−g)(1)=7
left parenthesis f minus g right parenthesis left parenthesis 1 right parenthesis equals 7
(f−g)(1)=−7
left parenthesis f minus g right parenthesis left parenthesis 1 right parenthesis equals negative 7
(f−g)(1)=−1
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Scarlett sells designer ponchos. The revenue, R, she makes from selling a batch of n ponchos is given by the function R(n)=20n, and the expenses, E, that she incurs from producing a batch of n ponchos is given by the function E(n)=10n√. Explain whether Scarlett earns a profit from producing and selling a batch of 9 ponchos.(1 point)
Responses
Scarlett earns a profit because R(10)E(10)=6.
Scarlett earns a profit because , Start Fraction upper r left parenthesis 10 right parenthesis over upper e left parenthesis 10 right parenthesis End Fraction equals 6, .
Scarlett earns a profit because R(10)−E(10)=150.
Scarlett earns a profit because , upper r left parenthesis 10 right parenthesis minus upper e left parenthesis 10 right parenthesis equals 150, .
Scarlett earns a profit because R(10)+E(10)=210.
Scarlett earns a profit because , upper r left parenthesis 10 right parenthesis plus upper e left parenthesis 10 right parenthesis equals 210, .
Scarlett does not earn a profit because E(10)−R(10)=−150.
To determine which statement is always true for any two functions f and g, we can analyze each statement and its implications.
Statement 1: If x is in the domain of f, then x must be in the domain of f−g.
Explanation: This statement is not always true. The domain of f−g depends on the individual domains of f and g, as well as their subtraction. It is possible for x to be in the domain of f but not in the domain of f−g if the subtraction of the two functions results in a domain restriction.
Statement 3: If x is not in the domain of fg, then x is not in the domain of g.
Explanation: This statement is not always true. The domain of fg depends on the individual domains of f and g, as well as their multiplication. It is possible for x to be not in the domain of fg but still in the domain of g if the multiplication of the two functions results in a domain restriction.
Statement 5: If x1 is in the domain of f and x2 is in the domain of g, then the product, x1x2, must be in the domain of fg.
Explanation: This statement is always true. If x1 is in the domain of f and x2 is in the domain of g, then the product x1x2 will not lead to any domain restrictions since multiplication doesn't impose domain limitations. Therefore, x1x2 will be in the domain of fg.
Statement 7: If x is in the domain of f+g, then x must be in the domain of f and x must be in the domain of g.
Explanation: This statement is always true. For x to be in the domain of f+g, both f(x) and g(x) must be defined at x. This implies that x is in the domain of f and x is in the domain of g.
From analyzing the statements, we can conclude that Statement 7 is the only statement that is always true.
Now, let's find (f−g)(x) for the given options:
Option 2: It is the difference of two functions, f(x)−g(x) times x.
This option incorrectly states that (f−g)(x) is multiplied by x. The correct answer is the difference of two functions, f(x)−g(x).
Lastly, in the question regarding finding fg and explaining the domain:
Option 1: The function is (fg)(x)=2x+9. The domain of fg is all real numbers because the domain of f and the domain of g are both all real numbers.
This option correctly states the function fg(x) and explains that the domain of fg is all real numbers since both f and g have domains that include all real numbers.
I hope this explanation helps clarify the answers and how to arrive at them! If you have any further questions, feel free to ask.