Solve the two sided inequality and show the solution on real line

7 < 1-2x ≤ 10

Given two functions as:
f(x) = and g(x) =
Find fog(x) also find the domain of f, g and fog

Simplify, then apply the rules of limit to evaluate

Given two functions as:

f(x) = x2-x-1 and g(x) =3/x
Find fog(x) also find the domain of f, g and fog

To solve the two-sided inequality 7 < 1-2x ≤ 10, we need to break it down into two separate inequalities and solve each one individually.

First, let's solve the left inequality, 7 < 1 - 2x.
Subtracting 1 from both sides, we get:
6 < -2x.
Dividing both sides by -2 (note that when dividing by a negative number, the inequality direction flips), we get:
-3 > x.

Next, let's solve the right inequality, 1-2x ≤ 10.
Subtracting 1 from both sides, we get:
-2x ≤ 9.
Dividing both sides by -2 (again, flipping the inequality direction), we get:
x ≥ -4.5.

So, combining the solutions of the two inequalities, we have:
-3 > x ≥ -4.5.

To show the solution on a real number line, draw a number line and label -3 and -4.5 on it. Then, shade the region between these two points to represent the solution.

Now, let's find fog(x) where f(x) and g(x) are given as functions. The composition fog(x) means plugging g(x) into f(x).

f(x) = ?
g(x) = ?

Unfortunately, the functions f(x) and g(x) are not provided in the question. To find fog(x), we need to know the expressions for f(x) and g(x). Please provide the functions' expressions, and I will help you find fog(x).

Moving on, to find the domain of a function, we need to identify all the possible values of x for which the function is defined.

The domain of f(x) is the set of all possible input values of x that yield valid outputs for the function. To find the domain, you need to consider any restrictions on x within the function's expression. For example, if there is a square root or a fraction with a denominator that can't be zero, those will impose restrictions on the domain.

Similarly, the domain of g(x) is the set of all possible input values of x that yield valid outputs for the function g(x). Consider any restrictions on x within the expression of g(x) to determine the domain.

Once we have the expressions for f(x) and g(x), we can find fog(x) by replacing x in f(x) with g(x). This means plugging in the expression of g(x) into the expression of f(x).

The domain of fog(x) will be determined by any restrictions on x in the compositions of f(x) and g(x). By analyzing the restrictions in each function, we can determine which values of x are valid inputs for fog(x) and hence find its domain.

Finally, to simplify an expression and evaluate the limit, please provide the specific expression or limit that needs to be simplified, so I can help you further.