The graph of f(x), a trigonometric function, and the graph of g(x) = c intersect at n points over the interval 0 <= x <= 2pi. There are m algebraic solutions to the equation f(x) = g(x), where m > n. Which of the following functions are most likely to be f(x)?
f(x) = cos x
f(x) = sqrt(sin x)
f(x) = cos^2 x + cos x
f(x) = sin^2 x + cos^2 x
1. A [ cos^2 (x) - sqrt(3)*cos(x) - 1 = 0 ]
2. B [ f(x) = sqrt(sin x) ]
3. D [ 7.6 hours ]
100% ur welcome
dirsegard this - (tagging purposes):
1. The equation 2sin(x) + sqrt (3) cot(x) = sin(x) is partially solved below. Which of the following steps could be included in the completed solution?
3. A buoy floats on the surface of the water. The height in meters of the buoy t hours after midnight, relative to sea level, is given by the equation f(x) = 5cos[(pi/6)t] + 4. Estimate how many hours elapse between the first two times the buoy is exactly 6 m above sea level.
Well, it seems like we've got a math problem on our hands! Let's see if we can use a dash of humor to find the solution.
Now, we're looking for the most likely candidate for f(x) out of the given options.
Option 1, f(x) = cos x, might seem like a good choice, but remember, not all cosines are created equal.
Option 2, f(x) = √(sin x), is an interesting choice. Square rooting can be quite unpredictable! It's like unruly hair - you never know how it's going to turn out.
Option 3, f(x) = cos^2 x + cos x, now we're getting somewhere! Combining two cosines seems like a solid strategy. It's like adding two clown noses together - double the fun!
And finally, option 4, f(x) = sin^2 x + cos^2 x, well, this one is a classic. Remember, in math, age is just a variable, and sin^2 x + cos^2 x never gets old!
So, if we're going for the most likely candidate, option 3, f(x) = cos^2 x + cos x, seems to be the clown with the best chances of making us laugh – I mean, being the f(x) we're looking for!
To determine which of the given functions are most likely to be f(x), we need to analyze their properties and how they intersect with g(x) = c.
Let's go through each function:
1. f(x) = cos x
The cosine function oscillates between -1 and 1 over the interval 0 <= x <= 2pi. If c is within this range, there will be at least one intersection point. However, since m > n, there will be additional algebraic solutions. Therefore, this function is a possibility, but not the most likely.
2. f(x) = √(sin x)
The square root of the sine function will only have a positive value when sin x is positive. Thus, the function f(x) = √(sin x) will be non-zero when sin x is positive. If g(x) = c is positioned such that c is positive, there will be at least one intersection point. However, since m > n, there will be additional algebraic solutions. Therefore, this function is also a possibility, but not the most likely.
3. f(x) = cos^2 x + cos x
Rewriting cos^2 x + cos x as cos x (cos x + 1), we see that f(x) is composed of two terms, one being the cosine function. The cosine function oscillates between -1 and 1 over the interval 0 <= x <= 2pi. The additional term cos x + 1 takes on values between 0 and 2 over the same interval. Thus, c can fall within the range of this function for the given interval. Since m > n, this function is highly likely to be f(x).
4. f(x) = sin^2 x + cos^2 x
Using the Pythagorean identity sin^2 x + cos^2 x = 1, we arrive at f(x) = 1. Since g(x) = c is a constant, there will be no intersection points over the interval 0 <= x <= 2pi. Therefore, this function is not a likely candidate for f(x) since it does not intersect with g(x) = c.
In conclusion, the most likely functions to be f(x) are f(x) = cos x and f(x) = cos^2 x + cos x.
To determine which of the given functions f(x) is most likely to intersect g(x) at the desired number of points, we need to understand the nature of the functions and analyze their graphs.
Let's start by examining the given options:
1. f(x) = cos x
2. f(x) = √(sin x)
3. f(x) = cos^2 x + cos x
4. f(x) = sin^2 x + cos^2 x
Option 1: f(x) = cos x
The function f(x) = cos x is a periodic function with a range of [-1, 1]. Its graph oscillates between 1 and -1 as x changes. Therefore, its graph will intersect g(x) = c at multiple points for various values of c. This makes it a plausible option for f(x).
Option 2: f(x) = √(sin x)
The function f(x) = √(sin x) is a non-negative function due to the square root. Its graph will always be above or on the x-axis, and it ranges from 0 to 1. Since g(x) = c can take any real value, the graphs of f(x) and g(x) are unlikely to intersect multiple times over the given interval. Thus, it is less likely to be f(x).
Option 3: f(x) = cos^2 x + cos x
This function is a combination of two cosine functions. By rewriting it as f(x) = cos x (cos x + 1), we see that it will intersect g(x) at those x-values where cos x + 1 = c. As mentioned before, the cosine function can intersect g(x) at multiple points, so this option is a plausible choice.
Option 4: f(x) = sin^2 x + cos^2 x
This function simplifies to f(x) = 1. Regardless of the value of x or c, g(x) = c will never intersect f(x) = 1 more than once. Therefore, this option is unlikely to be the correct choice.
In summary, the functions f(x) = cos x and f(x) = cos^2 x + cos x are the most likely options to be f(x) because they allow for multiple intersections with g(x) = c over the interval 0 <= x <= 2pi.
Go to any number of web sites that do graphs. (a good one is wolframalpha.com)
Take a look at the functions.
For example, cosx intersects y=c at 1 or 2 points during one period, depending on c.
see http://www.wolframalpha.com/input/?i=cosx
for the last one, remember your most basic trig identity.