Given if f(sinx)+f(cosx)=x then find domain and range of f(x).
To find the domain and range of the function f(x), let's analyze the given equation f(sinx) + f(cosx) = x.
First, let's identify the domains of sin(x) and cos(x). Both sin(x) and cos(x) can take any real number as an input, so their domains are -∞ to +∞.
For f(x) to be defined, both f(sinx) and f(cosx) must be defined. Since the sum of two defined functions equals x, we can deduce that f(x) must be defined for all real numbers.
Therefore, the domain of f(x) is -∞ to +∞.
Now let's analyze the range of f(x). To do this, we need to understand the behavior of f(x) for different values of x.
From the given equation, we can observe that f(sinx) + f(cosx) = x. Since the domain of sin(x) and cos(x) is -∞ to +∞, the values of sin(x) and cos(x) can vary from -1 to +1.
Let's express sin(x) and cos(x) as y:
sin(x) = y, -1 ≤ y ≤ 1
cos(x) = y, -1 ≤ y ≤ 1
Now, the equation becomes:
f(y) + f(y) = x
From this equation, we can see that the range of f(x) depends on the range of x. As x can vary from -∞ to +∞, the range of f(x) can also vary from -∞ to +∞.
Therefore, the range of f(x) is also -∞ to +∞.
In summary:
- The domain of f(x) is -∞ to +∞.
- The range of f(x) is -∞ to +∞.