Find left parenthesis lower f circle lower g right parenthesis left parenthesis x right parenthesis where lower f left parenthesis x right parenthesis equals Start Fraction 1 over x plus 1 End Fraction and lower g left parenthesis x right parenthesis equals negative Start Fraction 1 over x End Fraction.
The left parenthesis lower f circle lower g right parenthesis left parenthesis x right parenthesis is given by:
(lower f ∘ lower g)(x) = lower f(lower g(x))
First, we need to find lower f(left parenthesis x right parenthesis):
lower f(left parenthesis x right parenthesis) = 1/(x + 1)
Next, we need to find lower g(left parenthesis x right parenthesis):
lower g(left parenthesis x right parenthesis) = -1/x
Now, we can find lower f(lower g(x)):
lower f(lower g(x)) = 1/((lower g(x)) + 1)
= 1/((-1/x) + 1)
= 1/(1 - 1/x)
= 1/((x - 1)/x)
= x/(x - 1)
Therefore, the expression (lower f ∘ lower g)(x) equals x/(x - 1).
Find the domain of left parenthesis Start Fractionlower f over lower g End Fraction right parenthesis left parenthesis x right parenthesis where lower f left parenthesis x right parenthesis equals Start Root x plus 2 End Root and lower g left parenthesis x right parenthesis equals Start Root 3 minus x End Root.
To find the domain of the expression (lower f / lower g)(x), we need to determine the values of x for which both lower f(x) and lower g(x) are defined.
Given: lower f(x) = √(x + 2) and lower g(x) = √(3 - x)
For lower f(x) to be defined, the expression inside the square root (x + 2) must be non-negative:
x + 2 ≥ 0
x ≥ -2
For lower g(x) to be defined, the expression inside the square root (3 - x) must be non-negative:
3 - x ≥ 0
-x ≥ -3
x ≤ 3
Combining the two restrictions, we find the common domain:
-2 ≤ x ≤ 3
Therefore, the domain of (lower f / lower g)(x) is [-2, 3].
that answer is wrong
Apologies for the incorrect response.
To find the domain of the expression (lower f / lower g)(x), we need to consider two conditions:
1. The denominator, lower g(x) ≠ 0, since division by zero is undefined.
2. The radicand inside any square root, including both lower f(x) and lower g(x), must be non-negative.
Given: lower f(x) = √(x + 2) and lower g(x) = √(3 - x)
Now let's determine the values of x that satisfy the above conditions:
1. The denominator lower g(x) must be non-zero:
3 - x ≠ 0
x ≠ 3
2. The radicands of both lower f(x) and lower g(x) must be non-negative:
For lower f(x): x + 2 ≥ 0, which is true for all x ≥ -2.
For lower g(x): 3 - x ≥ 0, which is true for all x ≤ 3.
Combining these conditions, the common domain of (lower f / lower g)(x) is given by:
-2 ≤ x < 3
Therefore, the domain of the expression (lower f / lower g)(x) is [-2, 3).
Determine if lower f left parenthesis x right parenthesis equals Start Fraction 4 x plus 1 over x minus 5 End Fraction and lower g left parenthesis x right parenthesis equals Start Fraction 5 x minus 1 over x plus 4 End Fraction are inverses of each other. If they are not inverses, explain why.
To determine if lower f(x) = (4x + 1)/(x - 5) and lower g(x) = (5x - 1)/(x + 4) are inverses of each other, we need to check if their compositions satisfy the following conditions:
1. lower f ∘ lower g(x) = x, for all x in their common domain.
2. lower g ∘ lower f(x) = x, for all x in their common domain.
First, let's find the composition lower f ∘ lower g(x):
lower f ∘ lower g(x) = lower f(lower g(x))
Substituting the expression for lower g(x) into lower f(x):
lower f ∘ lower g(x) = lower f((5x - 1)/(x + 4))
Simplifying lower f((5x - 1)/(x + 4)):
lower f ∘ lower g(x) = (4((5x - 1)/(x + 4)) + 1)/((5x - 1)/(x + 4) - 5)
= (20x - 4 + x + 4)/((5x - 1) - 5(x + 4))
= (21x)/(-24x - 21)
Now let's find the composition lower g ∘ lower f(x):
lower g ∘ lower f(x) = lower g(lower f(x))
Substituting the expression for lower f(x) into lower g(x):
lower g ∘ lower f(x) = lower g((4x + 1)/(x - 5))
Simplifying lower g((4x + 1)/(x - 5)):
lower g ∘ lower f(x) = (5((4x + 1)/(x - 5)) - 1)/((4x + 1)/(x - 5) + 4)
= (20x + 5 - 1)/((4x + 1) + 4(x - 5))
= (20x)/(-4x - 19)
Now, we need to check if both compositions are equal to x.
For lower f ∘ lower g(x) = x:
(21x)/(-24x - 21) = x
Cross-multiplying:
21x = -24x^2 - 21x
24x^2 + 42x = 0
6x(x + 7) = 0
This equation has two solutions: x = 0 and x = -7.
For lower g ∘ lower f(x) = x:
(20x)/(-4x - 19) = x
Cross-multiplying:
20x = -4x^2 - 19x
4x^2 + 39x = 0
x(4x + 39) = 0
This equation has two solutions: x = 0 and x = -39/4.
Both compositions have a common solution of x = 0.
Therefore, lower f(x) = (4x + 1)/(x - 5) and lower g(x) = (5x - 1)/(x + 4) are inverses of each other, except for the case of x = -7 and x = -39/4, which are excluded from the domain due to denominator restrictions.
options
They are not inverses because left parenthesis lower f circle lower g right parenthesis left parenthesis x right parenthesis equals negative x.
They are inverses.
They are not inverses because left parenthesis lower f circle lower g right parenthesis left parenthesis x right parenthesis equals x.
They are not inverses because left parenthesislower f circle lower g right parenthesis left parenthesis x right parenthesis equals Start Fraction 1 over x End Fraction
The correct answer is:
They are not inverses because (lower f ∘ lower g)(x) = -x.