for the given function f&g, find the following and state the domain of each result
f(x)=2x+1/9x-5; g(x)4x/9x-5
a.(f+g)(x)=
what is the domain?
b.(f/g)(x)=
what is the domain?
please show work
answer the following for the given quadratic function
f(x)=x^2+10x-11
a. what is the vertex (h,k) of f?
b. what is the axis of symmetry?
please show work
1. is answered below on related links
2.
f(x) = x^2+10X+25 - 11 - 25
f(X) = (X+5)^2 - 36
now read off the vertex, which lies on the axis of symmetry.
a. To find (f+g)(x), we need to add f(x) and g(x):
(f+g)(x) = f(x) + g(x) = (2x+1)/(9x-5) + (4x)/(9x-5)
To simplify this expression, we need to find a common denominator:
(f+g)(x) = (2x+1+4x)/(9x-5)
Combining like terms, we get:
(f+g)(x) = (6x+1)/(9x-5)
The domain is the set of all real numbers except for values that make the denominator zero. In this case, we cannot have 9x-5=0 because it would result in division by zero. So we solve the equation 9x-5=0:
9x = 5
x = 5/9
Therefore, the domain of (f+g)(x) is all real numbers except x = 5/9.
b. To find (f/g)(x), we need to divide f(x) by g(x):
(f/g)(x) = f(x) / g(x) = (2x+1)/(9x-5) / (4x)/(9x-5)
Dividing by a fraction is the same as multiplying by its reciprocal:
(f/g)(x) = (2x+1)/(9x-5) * (9x-5)/(4x)
Simplifying, we get:
(f/g)(x) = (2x+1)/(4x)
The domain is the set of all real numbers except for values that make the denominator zero. In this case, we cannot have 4x=0 because it would result in division by zero. So we solve the equation 4x=0:
x = 0
Therefore, the domain of (f/g)(x) is all real numbers except x = 0.
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For the quadratic function f(x) = x^2 + 10x - 11:
a. To find the vertex (h,k), we can use the formula h = -b/2a and substituting it back into the quadratic equation to find k.
In this case, a = 1, b = 10, and c = -11:
h = -10/(2*1) = -10/2 = -5
Substituting h = -5 back into the equation:
k = (-5)^2 + 10*(-5) - 11 = 25 - 50 - 11 = -36
Therefore, the vertex of f(x) is (-5, -36).
b. The axis of symmetry is a vertical line that passes through the vertex, so its equation can be found using the x-coordinate of the vertex.
In this case, the x-coordinate of the vertex is -5.
Therefore, the axis of symmetry is x = -5.