An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 100100 engines and the mean pressure was 4.94.9 pounds/square inch (psi). Assume the population standard deviation is 0.60.6. The engineer designed the valve such that it would produce a mean pressure of 4.84.8 psi. It is believed that the valve does not perform to the specifications. A level of significance of 0.020.02 will be used. Find the P-value of the test statistic. Round your answer to four decimal places.

C)  I, I and III

In other words, statements I through III are all true.

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Explanation:

Apply substitution to say that,

We can see that the cube root and cube exponents cancel each other out, and everything simplifies to simply x.

Now compute the other composite function, where g(x) is the outer function this time.

Again, the cubing and cube root cancel each other out.

Since both f(g(x)) = x and g(f(x)) = x are true equations, this means that functions f(x) and g(x) are inverses of each other.

The two functions undo one another.

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As a simpler example, consider these two functions:

f(x) = x+5

g(x) = x-5

The f(x) function adds 5 to whatever the input is, and the g(x) goes in the opposite direction to subtract 5 to undo what f(x) did. For this example, you should find that f(g(x)) = x and g(f(x)) = x

Applying the composite of these inverses means that the final output is whatever the original input was. If we start with x = 87, then x+5 = 87+5 = 92, and then 92-5 = 87 which was the original input. Overall, no change has happened.