Suppose a railroad rail is 5 kilometers and it expands on a hot day by 18 centimeters in length. Approximately how many meters w
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Hello!
To prove that f(x) = 2x - 1 and g(x) = x/2 + 1/2, we can use a composite function. Composite functions are basically (f ∘ g)(x). It combines two functions into one. If they are true inverses, then the answer must be equal to x.
(f ∘ g)(x) = 2(x/2 + 1/2) - 1
(f ∘ g)(x) = x + 1 - 1
(f ∘ g)(x) = x
(g ∘ f)(x) = (2x - 1)/2 + 1/2
(g ∘ f)(x) = x - 1/2 + 1/2
(g ∘ f)(x) = x
Since (g ∘ f)(x) and (f ∘ g)(x) are both equal to x, then the functions of f(x) and g(x) are inverses of each other.
Also, in order for two functions to be inverses, these two functions need to be reflected over the line y = x. In the graph shown below, y = x is in red, y = 2x - 1 is blue, and y = x/2 + 1/2 is green. Looking the graph, you can see they are reflected over the line y = x.
Therefore, the function f(x) = 2x - 1 and g(x) = x/2 + 1/2 are true inverses of each other.
Answer:
n=206
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The population proportion have the following distribution
Solution to the problem
In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 99% of confidence, our significance level would be given by and
. And the critical value would be given by:
The margin of error for the proportion interval is given by this formula:
(a)
And on this case we have that , we can use as prior estimate of p 0.5, since we don't have any other info provided, and we are interested in order to find the value of n, if we solve n from equation (a) we got:
(b)
And replacing into equation (b) the values from part a we got:
And rounded up we have that n=206