I. Fill in each blank with the correct entry. Pick your answer from the box. one-to-one...

  

Transcribed Image Text:

I. Fill in each blank with the correct entry. Pick your answer from the box. one-to-one angles -1<x<1 one-to-many values -1<y<1 many-to-one vertical line test sin y = x range horizontal line test sys sin x = y domain periodic inverse sine function sine of inverse function We recall that for a function to have an inverse, it must be (1). every value in its domain must correspond to only one value in its (2), An example of which is f(x) = x +1. That means ,and vice versa. Suppose we input x = 3. We will get only one output, which is 4; likewise, if we want the value of f(x) to be 4, we can think of only one x-value that will give that – and that is 3. Hence, there exists a one-to-one relation between x and f(x). Meanwhile, in g(x) = |x + 11, the relation between x and g(x) is not one-to-one. Suppose we input x = 3. We will surely get only one output, which is 4; however, besides 3 we can also use another number to get the same output - and that number is -5. Ergo, such and does not have an inverse. function is (3). The trigonometric function f(x) = sin x is also many-to-one. Remember that our and the sine of some angles are equal. For example, the sine - Another way to prove that it is not one-to-one is by It is obvious that its graph will fail because the inputs here are (4). values of and are both (5). subjecting its graph to the (6). line will intersect it at more than one point since it is (7). But, what if we the take only a part of its graph so that no value in the (8). is repeated? It is like restricting its (9). of its graph where x is from - to,would it still fail the test? Not anymore. to a smaller set. Say we only take the part Therefore, by defining the domain of y = sin x to be -<xs,we successfully turn it into a one-to-one function whose range is –1<y< 1; thus, we can now find its inverse, which we will simply call (10). We then invent the symbol below to denote the latter: y = sin-x or y = arcsin x. We read it as "y is the inverse of sine of x" or "y is the arcsine of x", and its domain is As the name suggests, our input now is a value (11) i range is (12) while our output is an angle. For example, given that x =and you're asked to solve for y, it's like you are asked to find the angle within (14). whose sine value is 2. Finally, earmark that y = sin x and (15). are equivalent. We will need this fact in our future lessons. I. Fill in each blank with the correct entry. Pick your answer from the box. one-to-one angles -1<x<1 one-to-many values -1<y<1 many-to-one vertical line test sin y = x range horizontal line test sys sin x = y domain periodic inverse sine function sine of inverse function We recall that for a function to have an inverse, it must be (1). every value in its domain must correspond to only one value in its (2), An example of which is f(x) = x +1. That means ,and vice versa. Suppose we input x = 3. We will get only one output, which is 4; likewise, if we want the value of f(x) to be 4, we can think of only one x-value that will give that – and that is 3. Hence, there exists a one-to-one relation between x and f(x). Meanwhile, in g(x) = |x + 11, the relation between x and g(x) is not one-to-one. Suppose we input x = 3. We will surely get only one output, which is 4; however, besides 3 we can also use another number to get the same output - and that number is -5. Ergo, such and does not have an inverse. function is (3). The trigonometric function f(x) = sin x is also many-to-one. Remember that our and the sine of some angles are equal. For example, the sine - Another way to prove that it is not one-to-one is by It is obvious that its graph will fail because the inputs here are (4). values of and are both (5). subjecting its graph to the (6). line will intersect it at more than one point since it is (7). But, what if we the take only a part of its graph so that no value in the (8). is repeated? It is like restricting its (9). of its graph where x is from - to,would it still fail the test? Not anymore. to a smaller set. Say we only take the part Therefore, by defining the domain of y = sin x to be -<xs,we successfully turn it into a one-to-one function whose range is –1<y< 1; thus, we can now find its inverse, which we will simply call (10). We then invent the symbol below to denote the latter: y = sin-x or y = arcsin x. We read it as "y is the inverse of sine of x" or "y is the arcsine of x", and its domain is As the name suggests, our input now is a value (11) i range is (12) while our output is an angle. For example, given that x =and you're asked to solve for y, it's like you are asked to find the angle within (14). whose sine value is 2. Finally, earmark that y = sin x and (15). are equivalent. We will need this fact in our future lessons.

Answer rating: 100% (QA)

Answe... View the full answer