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

7 < 1-2x ≤ 10

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

7 < 1-2x ≤ 10

6 < -2x ≤ 9
-3 > x ≥ -9/2
or in the more usual form ..

-9/2 ≤ x < -3

the second part of your question I had already answered for you yesterday, but here it is again ...

domain of f : any real number
domain of g : any real number except x ≠ 0

fog
= f(g(x))
= f(3/x)
= (3/x)^2 - 3/x - 1

clearly the domain of fog is any real number, except x≠0

To solve the two-sided inequality 7 < 1 - 2x ≤ 10, we can break it down into two separate inequalities.

First, let's solve 7 < 1 - 2x:

7 < 1 - 2x
Subtract 1 from both sides:
6 < -2x
Divide both sides by -2 (remember to reverse the inequality when dividing by a negative number):
-3 > x

Next, let's solve 1 - 2x ≤ 10:

1 - 2x ≤ 10
Subtract 1 from both sides:
-2x ≤ 9
Divide both sides by -2 (again, remember to reverse the inequality):
x ≥ -4.5

So the solution to the two-sided inequality is -3 > x ≥ -4.5.

To show this solution on a real number line, draw a line with a point labeled -3 on the left side and a point labeled -4.5 on the right side. You can shade the area between these two points to represent the solution set, indicating that any value of x within this shaded region satisfies the inequality.

Now, let's find fog(x) for the given functions:

f(x) = x^2 - x - 1
g(x) = 3/x

To find fog(x), we substitute g(x) into f(x) by replacing the variable x in f(x) with g(x):

fog(x) = f(g(x))

In this case, g(x) is 3/x, so we substitute it into f(x):

fog(x) = f(3/x)

To evaluate f(3/x), we need to replace the x in f(x) with 3/x:

fog(x) = (3/x)^2 - (3/x) - 1

Now, we simplify the expression:

fog(x) = 9/x^2 - 3/x - 1

To find the domain of f(x), g(x), and fog(x), we need to consider any restrictions on the values of x that make the expressions undefined. In this case:

The domain of f(x) is all real numbers since there are no restrictions on x.

The domain of g(x) is all real numbers except x = 0 since dividing by zero is undefined.

The domain of fog(x) is the intersection of the domains of f(x) and g(x). Since the domain of f(x) is all real numbers and the domain of g(x) is all real numbers except x = 0, the domain of fog(x) is all real numbers except x = 0.