1. Using the definitions of odd and even functions, explain why y=sin x+1 is neither odd nor even.
2.Using the definition of an even function, show that y=-cos is even.
What is it you don't understand here. Apply the definition
1. To determine whether a function is odd or even, we need to understand the definitions of odd and even functions.
- An even function is symmetric with respect to the y-axis, meaning that the function satisfies the condition f(x) = f(-x) for all values of x. In other words, replacing x with -x in the function should not change its value.
- On the other hand, an odd function is symmetric with respect to the origin, meaning that the function satisfies the condition f(x) = -f(-x) for all values of x. In other words, replacing x with -x in the function should change its value by a negative sign.
Now, let's consider the function y = sin(x) + 1. To determine whether it is odd or even, we need to test whether it satisfies the conditions for even or odd functions.
First, let's test the condition for evenness: f(x) = f(-x)
If we substitute -x for x in the function, we get sin(-x) + 1. However, this is not equal to sin(x) + 1, as the sine function is not even. Therefore, the function does not satisfy the condition for evenness.
Next, let's test the condition for oddness: f(x) = -f(-x)
If we substitute -x for x in the function, we get sin(-x) + 1. However, this is not equal to -[sin(x) + 1]. Therefore, the function does not satisfy the condition for oddness.
Since the function y = sin(x) + 1 does not satisfy the conditions for evenness or oddness, we can conclude that it is neither an even nor an odd function.
2. To show that y = -cos(x) is an even function, we need to demonstrate that it satisfies the condition for evenness: f(x) = f(-x).
Let's substitute -x for x in the function:
f(-x) = -cos(-x)
Using the property of cosine function, cos(-x) = cos(x), so we can simplify the equation as follows:
f(-x) = -cos(x)
If we compare this to the original function f(x) = -cos(x), we see that f(-x) has the same value as f(x), which means that the function is symmetric with respect to the y-axis. Therefore, we can conclude that y = -cos(x) is an even function.