By what fraction of the period of f(x)=10 cos(7 x) has the graph of f(x) been shifted to obtain the graph of g(x)=10 cos(7x- pi/2). how do I approach this and solve for the fraction shift and the domain and range?

2 pi is a whole circle or period

pi/2 is a quarter of a circle or period

the domain of this function is all real x, in other words x can be any real number positive or negative
However cosine is between -1 and +1 so range is between -10 and +10

the period of y = 10cos(7x) is 2π/7

rewriting the second as
y = 10cos 7(x - π/14) we can see that the phase shift was
π/14 to the right.

so what fraction is that phase shift of the period?
(π/14)/(2π/7)
or
1/4 of the phase shift.

humm, looks like we agree in the end :)

Damon, when I opened the page and started to answer this question, I left it alone to watch the end of the WorldCup football game, and when I resumed I did not refresh the page, thus did not see your answer was already posted.

To find the fraction of the period by which the graph of f(x) has been shifted to obtain the graph of g(x), we need to compare the periods of f(x) and g(x). We can do this by examining the coefficient of x inside the cosine function.

In f(x) = 10 cos(7x), the coefficient of x is 7. The period of f(x) is given by T = 2π/7. This means that the graph of f(x) repeats itself every T = 2π/7 units.

In g(x) = 10 cos(7x - π/2), the coefficient of x is 7, which is the same as in f(x). However, there is an additional phase shift of -π/2 (or 90 degrees) inside the cosine function. This phase shift tells us how much the graph of g(x) has been shifted horizontally.

To find the fraction of the period shifted, we need to compare the phase shift to the period. Since the period is T = 2π/7, and the phase shift is -π/2, we can express the phase shift in terms of the fraction of the period as follows:

Fraction Shift = (Phase Shift) / (Period)
= (-π/2) / (2π/7)
= -π/2 * 7/2π
= -7/4

So, the graph of g(x) has been shifted by a fraction of -7/4 of the period of f(x).

Regarding the domain and range:

The domain of both f(x) and g(x) is the set of all real numbers since the cosine function is defined for all real values of x.

The range of the cosine function is -1 ≤ cos(x) ≤ 1. Therefore, both f(x) and g(x) have a range of -10 ≤ f(x), g(x) ≤ 10, given the amplitude of 10 multiplied by the cosine function.

I hope this helps! Let me know if you have any further questions.