5. If a trig equation has one answer, how many answers will it have? (Provided you do not restrict the domain?)
6. Why are the graphs of the inverse functions restricted to one period of the function?
I only need to answer these two to finish my homework and I can't figure out what to do, can someone help me? Thank you!
#5 since trig functions are periodic, they have infinitely many solutions.
#6 So the function and its inverse are 1-to-1. All the trig functions fail the horizontal-line test for inverses unless the domain is restricted, just like something simple like y=x^2.
5. If a trig equation has one answer, it will typically have infinitely many answers. This is because trigonometric functions are periodic, meaning they repeat their values in a regular pattern. For example, the sine function has a period of 2π, which means its values repeat every 2π radians or 360 degrees. So, if an equation has one solution within a certain range of values, it will have infinitely many solutions that repeat at regular intervals throughout the entire domain.
6. The graphs of inverse functions are restricted to one period of the function because they need to pass the horizontal line test to be considered functions. The horizontal line test states that a function must pass the test if no horizontal line intersects the graph of the function more than once. If we were to graph the inverse function over multiple periods, it would fail the horizontal line test because multiple horizontal lines can intersect the graph at different points within each period. To ensure that the inverse function is a valid function, we restrict it to one period of the original function.
Of course! I would be happy to help you with these questions.
5. If a trigonometric equation has one answer, it means that there is only one value of the variable that satisfies the equation. However, if we do not restrict the domain (or range) of the equation, then there will be infinitely many solutions or answers.
To find the number of solutions or answers to a trigonometric equation, we often consider the period of the trigonometric function involved in the equation. A period is the shortest interval or range of the function in which it repeats its values. For example, the sine function has a period of 2π, which means it will repeat its values every 2π units.
If we don't restrict the domain, the trigonometric equation will have infinitely many solutions, as the function will repeat its values infinitely often. In other words, we can find multiple solutions by adding or subtracting multiples of the period value from the original solution.
6. The reason why the graphs of inverse functions are often restricted to one period of the original function is to ensure that the inverse function is well-defined and has a one-to-one correspondence between its domain and range.
When we find the inverse of a function, we interchange the role of the x and y variables. However, for trigonometric functions, which are periodic, this interchange is not always straightforward as it would result in multiple values.
By restricting the domain of the inverse function to one period of the original function, we ensure that the inverse function is single-valued and possesses a one-to-one correspondence. This allows us to have a unique mapping between the input and output values, which is necessary for a well-defined inverse function.
So, in summary, the restriction of the graphs of inverse functions to one period of the original function is done to maintain a one-to-one correspondence and ensure a valid inverse function.