In Exercises 19–32, find the (a) domain and (b) range.

Question

𝔂 = 3 cos x + 4 sin x (Hint: A trig identity is required.)

Accepted Answer

Welcome back, everyone. What is the domain and range of the function G of X equals 5 cosine of x minus 3 X. So let's begin by analyzing the domain of the given function. Now, the domain of the function essentially corresponds to all of the X values for which our function is defined. We can see that our function consists of two trigonometric functions, sine and cosine. So now what we want to do is simply analyze cosine to begin with, cosine of X, and we can say that X belongs to all real numbers. We can take cosine off any value of X that we want. We have to recall that cosine is an infinite periodic oscillating function, right? And similarly, if we analyze sin of x. X belongs to all real numbers. We can take sign off any X that we want. Once again, it is an infinite oscillating periodic function. So if we simply multiply those by 5 and -3 respectively, well, this doesn't change anything because these coefficients only change the y range, right, but it doesn't have any effect on the domain. And now if we are subtracting these two functions, well, essentially it has no effect on the domain of the overall function, meaning the domain is X belongs to all real numbers. That's the domain of G of X. Now, when it comes to range, that's a bit more complicated because now we want to understand. What is the range of these functions? We're looking at the Y values. We're looking at the minimum and the maximum Y value. So essentially, if we get a periodic function, we want to identify the minimum. And the maximum. So what we're going to do is simply use one of the trigonometric identities. We have to recall that we can rewrite G of X. Which consists of a difference between a cosine function and sine function as. Are multiplied by cosine. Of X plus sum base angle alpha. Now the angle alpha is not so relevant because we're only interested in R. Now how do we express R? Well, essentially R is equal to square root of a 2 + b2 and now what are A and B? Well A is simply the coefficient adjacent to cosine, so it's 5. Our AS 5. And B is the coefficient adjacent to sine x and that's -3. And now if we evaluate R, this is equal to square root of 52 + -32. Which gives us 25 + 9, that's 34. So R is equal to square root of 34. And this means that we can rewrite G of X as square root of 34. Of cosine of x plus the phase angle alpha, which is irrelevant for the range, right, recall that our range is only defined by the amplitude. So what he wants to do is simply recall that a cosine function oscillates between -1 and 1. And in this case, well, we can multiply our inequality by square root of 34, right? Or simplicity To make it really accurate, we're going to see cosine of X plus alpha. It doesn't matter if we take X or alpha, right? The domain has no difference on our range. And now if we multiply all sides by square of 34, we're going to get negative, square root of 34 is less than or equal to square root of 34, cosine of x plus alpha, which is less than or equal to square root of 34. And this is where we have our G of X. So we can see that the lower bound is 2 of 34 and the upper bound is square root of 34. And this is how we get our range. So now let's conclude the findings, starting with the domain X belongs to all real numbers or from negative infinity up to positive infinity, and for our range. Y belongs to the minimum value is negative square root of 34 inclusive. And the maximum value is square root of 34 inclusive. That's it. We have identified the final answers to this problem. Thank you for watching.

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = 3 cos x + 4 sin x (Hint: A trig identity is required.)

Key Concepts

Domain of a Function

Range of a Function

Trigonometric Identities

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