Question
Find the inverse of the function f left parenthesis x right parenthesis equals start fraction 7 x minus 4 over x plus 3 end fraction.
Answers
Answered by
GPT 3.5
To find the inverse of a function, we need to swap the roles of x and y and solve for y.
Let's write the original function as y = (7x - 4)/(x + 3).
Now, swap x and y: x = (7y - 4)/(y + 3).
To solve for y, let's multiply both sides by (y + 3):
x(y + 3) = 7y - 4.
Distribute on the left side:
xy + 3x = 7y - 4.
Move all the y terms to one side:
3x + 4 = 7y - xy.
Factor out y on the right side:
3x + 4 = y(7 - x).
Now, divide both sides by (7 - x):
(3x + 4)/(7 - x) = y.
Therefore, the inverse function is y = (3x + 4)/(7 - x).
Let's write the original function as y = (7x - 4)/(x + 3).
Now, swap x and y: x = (7y - 4)/(y + 3).
To solve for y, let's multiply both sides by (y + 3):
x(y + 3) = 7y - 4.
Distribute on the left side:
xy + 3x = 7y - 4.
Move all the y terms to one side:
3x + 4 = 7y - xy.
Factor out y on the right side:
3x + 4 = y(7 - x).
Now, divide both sides by (7 - x):
(3x + 4)/(7 - x) = y.
Therefore, the inverse function is y = (3x + 4)/(7 - x).
Answer
options
a) This function is not one-to-one and therefore has no inverse.
b) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction
Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction
c) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative x plus 3 over 7 x minus 4 end fraction
Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative x plus 3 over 7 x minus 4 end fraction
d) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction 3 x plus 4 over x plus 7 end fraction
a) This function is not one-to-one and therefore has no inverse.
b) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction
Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction
c) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative x plus 3 over 7 x minus 4 end fraction
Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative x plus 3 over 7 x minus 4 end fraction
d) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction 3 x plus 4 over x plus 7 end fraction
Answered by
GPT 3.5
The correct option is:
b) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction
b) f superscript negative 1 baseline left parenthesis x right parenthesis equals start fraction negative 3 x minus 4 over x minus 7 end fraction
Answer
Determine the domain of left parenthesis start fraction f over g end fraction right parenthesis left parenthesis x right parenthesis when f left parenthesis x right parenthesis equals start fraction 1 over x end fraction and g left parenthesis x right parenthesis equals start root x plus 8 end root.
Answered by
GPT 3.5
To determine the domain of the expression f/g, we need to find the values of x for which the denominator g(x) is not equal to 0.
The function g(x) = √(x + 8) is defined for all values of x that make the expression under the square root non-negative. Therefore, we need to solve the inequality:
x + 8 ≥ 0
Subtracting 8 from both sides, we have:
x ≥ -8
So, the domain of g(x) is x ≥ -8.
Now, let's consider the function f(x) = 1/x. The domain of f(x) is all real numbers except 0 because division by 0 is undefined.
Finally, for the expression f/g, we need to consider the common domain of f(x) and g(x). That means we need to find the intersection of the domains of f(x) and g(x).
Since the domain of g(x) is x ≥ -8 and the domain of f(x) is all real numbers except 0, the common domain is the intersection of these two sets, which is:
x ≥ -8 and x ≠ 0
In interval notation, the domain of f/g is (-∞, -8) U (-8, 0) U (0, ∞).
The function g(x) = √(x + 8) is defined for all values of x that make the expression under the square root non-negative. Therefore, we need to solve the inequality:
x + 8 ≥ 0
Subtracting 8 from both sides, we have:
x ≥ -8
So, the domain of g(x) is x ≥ -8.
Now, let's consider the function f(x) = 1/x. The domain of f(x) is all real numbers except 0 because division by 0 is undefined.
Finally, for the expression f/g, we need to consider the common domain of f(x) and g(x). That means we need to find the intersection of the domains of f(x) and g(x).
Since the domain of g(x) is x ≥ -8 and the domain of f(x) is all real numbers except 0, the common domain is the intersection of these two sets, which is:
x ≥ -8 and x ≠ 0
In interval notation, the domain of f/g is (-∞, -8) U (-8, 0) U (0, ∞).
Answer
Determine the domain and range for the inverse of f left parenthesis x right parenthesis equals start fraction 1 over 4 x end fraction plus 2.
Answered by
GPT 3.5
To find the domain and range of the inverse of f(x), we need to first find the inverse of the function.
The function is given as f(x) = (1/(4x)) + 2.
To find the inverse of f(x), we swap x and y and solve for y:
x = (1/(4y)) + 2.
We can solve this equation for y by isolating the variable:
x - 2 = 1/(4y).
To eliminate the fraction, we can multiply both sides by 4y:
4yx - 8y = 1.
Rearranging the equation, we get:
4yx - 1 = 8y.
Now let's isolate y:
4yx - 8y = 1.
Factor out y from the left side:
y(4x - 8) = 1.
Now, divide both sides by (4x - 8):
y = 1/(4x - 8).
Therefore, the inverse of f(x) is given by g(x) = 1/(4x - 8).
Now, let's find the domain and range of the inverse function g(x):
The domain of g(x) is the set of x-values for which the function is defined. In this case, we cannot divide by zero, so we need to make sure that the denominator (4x - 8) is not equal to zero. Thus, we have:
4x - 8 ≠ 0.
Solving this inequality, we find:
4x ≠ 8,
x ≠ 8/4,
x ≠ 2.
Therefore, the domain of the inverse function is all real numbers except x = 2.
The range of the inverse function is the set of all possible y-values. Since the function g(x) = 1/(4x - 8) can take any real value depending on the input x (except when x = 2, as mentioned above), the range of the inverse function is all real numbers except y = 0.
In interval notation, the domain of the inverse function is (-∞, 2) U (2, ∞), and the range is (-∞, 0) U (0, ∞).
The function is given as f(x) = (1/(4x)) + 2.
To find the inverse of f(x), we swap x and y and solve for y:
x = (1/(4y)) + 2.
We can solve this equation for y by isolating the variable:
x - 2 = 1/(4y).
To eliminate the fraction, we can multiply both sides by 4y:
4yx - 8y = 1.
Rearranging the equation, we get:
4yx - 1 = 8y.
Now let's isolate y:
4yx - 8y = 1.
Factor out y from the left side:
y(4x - 8) = 1.
Now, divide both sides by (4x - 8):
y = 1/(4x - 8).
Therefore, the inverse of f(x) is given by g(x) = 1/(4x - 8).
Now, let's find the domain and range of the inverse function g(x):
The domain of g(x) is the set of x-values for which the function is defined. In this case, we cannot divide by zero, so we need to make sure that the denominator (4x - 8) is not equal to zero. Thus, we have:
4x - 8 ≠ 0.
Solving this inequality, we find:
4x ≠ 8,
x ≠ 8/4,
x ≠ 2.
Therefore, the domain of the inverse function is all real numbers except x = 2.
The range of the inverse function is the set of all possible y-values. Since the function g(x) = 1/(4x - 8) can take any real value depending on the input x (except when x = 2, as mentioned above), the range of the inverse function is all real numbers except y = 0.
In interval notation, the domain of the inverse function is (-∞, 2) U (2, ∞), and the range is (-∞, 0) U (0, ∞).
Answer
The function f left parenthesis x right parenthesis equals left parenthesis x minus 5 right parenthesis squared plus 2 is not one-to-one. Identify a restricted domain that makes the function one-to-one, and find the inverse function.
Answered by
GPT 3.5
To find a restricted domain that makes the function f(x) = (x - 5)^2 + 2 one-to-one, we need to restrict the values of x such that each input corresponds to a unique output.
To do this, we can restrict the domain to x ≤ 5 or x ≥ 5. Either restriction will make the function one-to-one because for any given value of x in the restricted domain, there will be only one corresponding value of f(x).
Let's choose the restricted domain x ≥ 5:
Domain: x ≥ 5
To find the inverse function, we can swap the roles of x and y and solve for y:
x = (y - 5)^2 + 2
To solve for y, let's isolate the term with the square:
x - 2 = (y - 5)^2
Taking the square root of both sides:
√(x - 2) = y - 5
Now, let's solve for y:
y = √(x - 2) + 5
Therefore, the inverse function is:
f^(-1)(x) = √(x - 2) + 5
To do this, we can restrict the domain to x ≤ 5 or x ≥ 5. Either restriction will make the function one-to-one because for any given value of x in the restricted domain, there will be only one corresponding value of f(x).
Let's choose the restricted domain x ≥ 5:
Domain: x ≥ 5
To find the inverse function, we can swap the roles of x and y and solve for y:
x = (y - 5)^2 + 2
To solve for y, let's isolate the term with the square:
x - 2 = (y - 5)^2
Taking the square root of both sides:
√(x - 2) = y - 5
Now, let's solve for y:
y = √(x - 2) + 5
Therefore, the inverse function is:
f^(-1)(x) = √(x - 2) + 5
Answer
options
a) restricted domain: x greater or equals 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 minus start root x minus 2 end root
restricted domain: Image with alt text: x greater or equals 5 ; Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 minus start root x minus 2 end root
b) restricted domain: x greater or equals 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x minus 2 end root
restricted domain: Image with alt text: x greater or equals 5 ; Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x minus 2 end root
c) restricted domain: x less than or equals 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x plus 2 end root
restricted domain: Image with alt text: x less than or equals 5 ; Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x plus 2 end root
d) restricted domain: x less than or equals negative 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x minus 2 end root
a) restricted domain: x greater or equals 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 minus start root x minus 2 end root
restricted domain: Image with alt text: x greater or equals 5 ; Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 minus start root x minus 2 end root
b) restricted domain: x greater or equals 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x minus 2 end root
restricted domain: Image with alt text: x greater or equals 5 ; Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x minus 2 end root
c) restricted domain: x less than or equals 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x plus 2 end root
restricted domain: Image with alt text: x less than or equals 5 ; Image with alt text: f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x plus 2 end root
d) restricted domain: x less than or equals negative 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 plus start root x minus 2 end root
Answered by
GPT 3.5
The correct option is:
a) restricted domain: x ≥ 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 minus start root x minus 2 end root
a) restricted domain: x ≥ 5; f superscript negative 1 baseline left parenthesis x right parenthesis equals 5 minus start root x minus 2 end root
Answer
Determine the domain of the function f left parenthesis x right parenthesis equals start fraction 1 over 5 x squared plus 23 x plus 12 end fraction.
Answered by
GPT 3.5
To determine the domain of the function f(x) = 1/(5x^2 + 23x + 12), we need to find the values of x for which the denominator is not equal to 0.
The denominator of the function is 5x^2 + 23x + 12. To find the values of x for which the denominator is 0, we can factorize the quadratic equation:
5x^2 + 23x + 12 = 0
Factoring the quadratic equation, we have:
(5x + 3)(x + 4) = 0
Setting each factor equal to 0, we get:
5x + 3 = 0 or x + 4 = 0
Solving each equation, we find:
x = -3/5 or x = -4
Therefore, the denominator is equal to 0 when x = -3/5 or x = -4.
To find the domain of the function, we exclude these values from the set of all real numbers (-∞, ∞).
Hence, the domain of the function f(x) is all real numbers except x = -3/5 and x = -4. In interval notation, the domain is (-∞, -4) U (-4, -3/5) U (-3/5, ∞).
The denominator of the function is 5x^2 + 23x + 12. To find the values of x for which the denominator is 0, we can factorize the quadratic equation:
5x^2 + 23x + 12 = 0
Factoring the quadratic equation, we have:
(5x + 3)(x + 4) = 0
Setting each factor equal to 0, we get:
5x + 3 = 0 or x + 4 = 0
Solving each equation, we find:
x = -3/5 or x = -4
Therefore, the denominator is equal to 0 when x = -3/5 or x = -4.
To find the domain of the function, we exclude these values from the set of all real numbers (-∞, ∞).
Hence, the domain of the function f(x) is all real numbers except x = -3/5 and x = -4. In interval notation, the domain is (-∞, -4) U (-4, -3/5) U (-3/5, ∞).
Answer
Find left parenthesis g circle f right parenthesis left parenthesis x right parenthesis when f left parenthesis x right parenthesis equals start root x plus 3 end root and g left parenthesis x right parenthesis equals start fraction x squared plus 2 over x end fraction.
Answered by
GPT 3.5
To find (g ∘ f)(x), we need to substitute the function f(x) into the function g(x) and simplify.
Given: f(x) = √(x + 3) and g(x) = (x^2 + 2) / x.
Substituting f(x) into g(x), we have:
(g ∘ f)(x) = g(f(x))
= g(√(x + 3))
= (√(x + 3))^2 + 2 / √(x + 3)
= (x + 3) + 2 / √(x + 3)
= x + 5 / √(x + 3)
Therefore, (g ∘ f)(x) = (x + 5) / √(x + 3).
Given: f(x) = √(x + 3) and g(x) = (x^2 + 2) / x.
Substituting f(x) into g(x), we have:
(g ∘ f)(x) = g(f(x))
= g(√(x + 3))
= (√(x + 3))^2 + 2 / √(x + 3)
= (x + 3) + 2 / √(x + 3)
= x + 5 / √(x + 3)
Therefore, (g ∘ f)(x) = (x + 5) / √(x + 3).
Answer
Find left parenthesis f circle g right parenthesis left parenthesis x right parenthesis and its domain when f left parenthesis x right parenthesis equals x squared plus 2 and g left parenthesis x right parenthesis equals start root x plus 5 end root.
Answered by
GPT 3.5
To find (f ∘ g)(x), we need to substitute the function g(x) into the function f(x) and simplify.
Given: f(x) = x^2 + 2 and g(x) = √(x + 5).
Substituting g(x) into f(x), we have:
(f ∘ g)(x) = f(g(x))
= f(√(x + 5))
= (√(x + 5))^2 + 2
= (x + 5) + 2
= x + 7
Therefore, (f ∘ g)(x) = x + 7.
The domain of (f ∘ g)(x) is the same as the domain of g(x), since g(x) is the input to f(x). The domain of g(x) is x + 5 ≥ 0, which means x ≥ -5.
Hence, the domain of (f ∘ g)(x) is x ≥ -5.
Given: f(x) = x^2 + 2 and g(x) = √(x + 5).
Substituting g(x) into f(x), we have:
(f ∘ g)(x) = f(g(x))
= f(√(x + 5))
= (√(x + 5))^2 + 2
= (x + 5) + 2
= x + 7
Therefore, (f ∘ g)(x) = x + 7.
The domain of (f ∘ g)(x) is the same as the domain of g(x), since g(x) is the input to f(x). The domain of g(x) is x + 5 ≥ 0, which means x ≥ -5.
Hence, the domain of (f ∘ g)(x) is x ≥ -5.
Answer
options
a) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals start root x squared plus 7 end root semicolon left bracket negative 5 comma infinity right parenthesisImage with alt text: left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals start root x squared plus 7 end root semicolon left bracket negative 5 comma infinity right parenthesis
b) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left bracket negative 5 comma infinity right parenthesis
Image with alt text: left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left bracket negative 5 comma infinity right parenthesis
c) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left parenthesis negative infinity comma infinity right parenthesis
Image with alt text: left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left parenthesis negative infinity comma infinity right parenthesis
d) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals start root x squared plus 7 end root semicolon left parenthesis negative infinity comma infinity right parenthesis
a) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals start root x squared plus 7 end root semicolon left bracket negative 5 comma infinity right parenthesisImage with alt text: left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals start root x squared plus 7 end root semicolon left bracket negative 5 comma infinity right parenthesis
b) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left bracket negative 5 comma infinity right parenthesis
Image with alt text: left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left bracket negative 5 comma infinity right parenthesis
c) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left parenthesis negative infinity comma infinity right parenthesis
Image with alt text: left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals x plus 7 semicolon left parenthesis negative infinity comma infinity right parenthesis
d) left parenthesis f circle g right parenthesis left parenthesis x right parenthesis equals start root x squared plus 7 end root semicolon left parenthesis negative infinity comma infinity right parenthesis
Answered by
GPT 3.5
The correct option is:
b) (f ∘ g)(x) = x + 7; domain: [-5, ∞)
b) (f ∘ g)(x) = x + 7; domain: [-5, ∞)