Four band members, harry, isabel, jose, and karen, are all boarding a bus to attend a concert. in each arrangement below, the first initial of each person’s name represents that person’s position as he or she boards the bus: first, second, third, and fourth. if we know that karen boards the bus first, which shows the possible outcomes for this event? khij khij, khji khij, khji, kjhi, kjih khij, khji, kjhi, kjih, kihj, kijh

Let's go through the choices one by one.

Choice A) The functions have the same range.

This is false.

The range of f(x) is the set of all positive numbers and 0, while g(x) has a range consisting of all negative numbers and 0. Compare the two graphs (see attache image below) to see the two ranges. The range is the set of possible y outputs of a function.

Choice B) The functions have the same domains.

This is false.

F(x) has a domain of positive numbers and 0 included as well. In sharp contrast, g(x) has a domain consisting of negative numbers and 0. The domain is the set of allowed x values of a function. Have a look at the graph to see how I got those domains.

Choice C) The only value that is in the domains of both functions is 0.

This is true.

Only x = 0 is allowed to be plugged into both functions. Something like x = 7 works for f(x), but isn't defined for g(x) (assuming we ignore complex numbers). Meanwhile, a value like x = -2 works for g(x) but not for f(x). Refer to choice B.

Choice D) There are no values that are in the ranges of both funcions

This is false.

Why? Because y = 0 is a possible output for both functions (plug x = 0 into each function). If we ignored 0, then the statement would be true and that there are no overlapping/shared y values between the two functions.

Choice E) The domain of g(x) is all values greater than or equal to 0.

This is false.

The function g(x) = -sqrt(-x) has a domain of x <= 0.

Choice F) The rainge of g(x) is all values less than or equal to 0.

This is true.

The graph of g(x) shows it extending forever downward so y becomes more and more negative. In other words, y can be any negative number or 0.

Final

Choice C and Choice F are true statements. The rest are false.