Let lower f left-parenthesis x right-parenthesis equals Start Fraction 4 x minus 3 over x minus 10 End Fraction and lower g left-parenthesis x right-parenthesis equals Start Fraction 2 x minus 8 over x minus 10 End Fraction. Find left-parenthesis f plus g right-parenthesis left-parenthesis x right-parenthesis. Assume all appropriate restrictions to the domain.
To find f(x) + g(x), we need to add the functions f(x) and g(x) together.
f(x) + g(x) = (4x - 3)/(x - 10) + (2x - 8)/(x - 10)
Since both terms have the same denominator, we can combine them into a single fraction:
f(x) + g(x) = (4x - 3 + 2x - 8)/(x - 10)
Simplifying the numerator:
f(x) + g(x) = (6x - 11)/(x - 10)
Therefore, f(x) + g(x) = (6x - 11)/(x - 10) with the restriction that x cannot equal 10 (due to the denominator).
Find the domain of the function extended-left-parenthesis Start Fraction f over g End Fraction extended-right-parenthesis left-parenthesis x right-parenthesis where lower f left-parenthesis x right-parenthesis equals x squared minus 9 and lower g left-parenthesis x right-parenthesis equals x squared minus 4 x plus 3
To find the domain of the function f/g(x), we need to determine the values of x for which the denominator g(x) is not equal to zero. This is because division by zero is undefined.
Given that g(x) = x^2 - 4x + 3, we set the denominator equal to zero and solve for x:
x^2 - 4x + 3 = 0
This equation can be factored as:
(x - 1)(x - 3) = 0
Setting each factor equal to zero gives us two possible values for x:
x - 1 = 0, which gives x = 1
x - 3 = 0, which gives x = 3
Therefore, for f/g(x) to be defined, x must not be equal to 1 or 3.
Thus, the domain of the function f/g(x) is all real numbers except x = 1 and x = 3.
choices
left-parenthesis negative infinity comma infinity right-parenthesisImage with alt text: left-parenthesis negative infinity comma infinity right-parenthesis
left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma infinity right-parenthesis
left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis
left-parenthesis negative infinity comma negative 3 right-parenthesis union left-parenthesis negative 3 comma negative 1 right-parenthesis union left-parenthesis negative 1 comma infinity right-parenthesis
The correct answer is:
(left-parenthesis negative infinity comma 1 right-parenthesis union left-parenthesis 1 comma 3 right-parenthesis union left-parenthesis 3 comma infinity right-parenthesis)
This is because the domain of the function f/g(x) is all real numbers except x = 1 and x = 3, so it includes all values from negative infinity to 1, excluding 1, then from 1 to 3, excluding 3, and finally all values greater than 3 up to positive infinity.
Let lower f left-parenthesis x right-parenthesis equals Start Root x minus 2 End Root and lower g left-parenthesis x right-parenthesis equals Start Root x plus 7 End Root. Find left-parenthesis f dot g right-parenthesis left-parenthesis x right-parenthesis. Assume all appropriate restrictions to the domain.
To find f(x)⋅g(x), we need to multiply f(x) and g(x) together.
f(x)⋅g(x) = (√(x - 2))⋅(√(x + 7))
When multiplying square roots, we can simplify the expression as follows:
f(x)⋅g(x) = √(x - 2)⋅√(x + 7) = √((x - 2)⋅(x + 7))
Expanding the expression inside the square root:
f(x)⋅g(x) = √(x^2 + 5x - 14)
Therefore, f(x)⋅g(x) = √(x^2 + 5x - 14) with the restriction that x^2 + 5x - 14 ≥ 0. This is because taking the square root of a negative number is undefined.
To determine the domain of the function, we need to find the values of x that satisfy the inequality x^2 + 5x - 14 ≥ 0.
Factoring the quadratic equation:
(x + 7)(x - 2) ≥ 0
Setting each factor greater than or equal to zero and solving:
x + 7 ≥ 0, which gives x ≥ -7
x - 2 ≥ 0, which gives x ≥ 2
So, the solution to the inequality x^2 + 5x - 14 ≥ 0 is x ≥ 2.
Therefore, the domain of the function f⋅g(x) is x ≥ 2.
choices
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 5 x minus 14Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 5 x minus 14
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 9 x minus 14
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 9 x minus 14
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 9 x minus 14 End Root
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 9 x minus 14 End Root
left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 5 x minus 14 End Root
The correct answer is:
(lower f dot lower g) (x) = x^2 + 5x - 14
This is because the product of f(x) and g(x) simplifies to x^2 + 5x - 14.