1. let f(x) = 4x-3/x-10 and g(x) = 2x-8/x-10. find (f+g)(x). assume all appropriate restrictions to the domain.
a) (f+g)(x) = 6x-11/x-10
b) (f+g)(x) = 2x+5/x-10
c) (f+g)(x) = 6x+11/x-10
d) (f+g)(x) = 6x-11/2x-20
2. find the domain of the function (f/g)(x) where f(x)=x²-9 and g(x)=x²-4x+3.
a) (-∞,∞)
b) (-∞,1) u (1, ∞)
c) (-∞,1) u (1,3) u (3, ∞)
d) (-∞,-3) u (-3,-1) u (-1, ∞)
3. let f(x) = √x-2 and g(x) = √x+7. find (f*g)(x). assume all appropriate restrictions to the domain.
a) (f*g)(x) = x^2+5x-14
b) (f*g)(x) = x^2+9x-14
c) (f*g)(x) = √x²+9x-14
d) (f*g)(x) = √x²+5x-14
4. determine the graph of (f-g)(x) when f(x) = 1/x and g(x) = √x.
5. find the range of (f+g)(x) when f(x)=(x+4)² and g(x)=3.
a) (-∞,∞)
b) [3, ∞)
c) (-∞, -3)
d) [-3, ∞)

The correct answers are:

(1) Option (a)

(2) Option (b)

(3) Option (d)

(4) Graph is attached with the answer along with the explanation (below)!

(5) Option (b)

Explanations:

(1) Given Data:

f(x) =

g(x) =

Required = (f+g)(x) = ?

The expression (f+g)(x) is nothing but the addition of f(x) and g(x). Therefore, in order to find (f+g)(x), we need to add both the given functions as follows:

Now we need to simplify the above equation as follows:

Hence the correct answer is Option (a)

(2) Given Data:

f(x) =

g(x) =

Before finding the domain of the expression , we need to first evalute that expression as follows:

Now we need to put the denominator equal to zero in order to know what values of x should not be in the domain of this function:

x-1 = 0

x = 1

It means that the domain of is all real numbers EXCEPT x = 1. The (closed) parentheses " ) " or "(" means that the number is not included in the domain. Therefore, we can write that the domain of is (Option b)

(3) Given Data:

f(x) =

g(x) =

Required = (f*g)(x) = ?

The expression (f*g)(x) is nothing but the multiplication of f(x) and g(x). Therefore, in order to find (f*g)(x), we need to multiply both the given functions as follows:

Now we need to simplify the above equation as follows:

(Option d)

(4) Given Data:

f(x) =

g(x) =

Required = The graph of (f-g)(x) = ?

Before plotting the graph let us evalute it first. (f-g)(x) is the subtraction of g(x) from f(x). Mathematically, we can write it as:

(f-g)(x) =

Now simplify:

Look at the graph attached with this answer. As you can see, at x=0, the graph shoots up! As at x=0, the value of function approaches to infinity.

(5) Given Data:

f(x) =

g(x) = 3

Required = Range of (f+g)(x) = ?

Before finding the range of (f+g)(x), we first need to write the function:

(f+g)(x) =

Now that we have written the function, the next step is to find the inverse of this function in order to obtain the range.To find the inverse, swap x with y, and y with x and put (f+g)(x) = y as follows:

(f+g)(x) = y =

Now swap:

x =

Now solve for y:

Take square-root on both sides:

As you know that the square root of negative numbers are the complex numbers, and in range, we do not include the complex numbers. Therefore, the values of x should be greater or equal to 3 to have the square-roots to be the real numbers. Therefore,

Range of (f+g)(x) = (Option b)

Note: "[" or "]" bracket is used to INCLUDE the value. It means that 3 is included in the range.

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

mark brainliest!