Prove that `f:R->R,f(x)=cosx` is a many-one into function. Change the domain and co-domain of `f` such that `f` become :

Prove that f:R rarr R,f(x)=cos x is a many one into function.Change the domain and co- domain of f such that f become:

Is the function f:RtoR defined f(x)=cos(2x+1) invertible? Give reasons. Can you modify domain and co-domain of f such that f becomes invertible?

Is the function f:RtoR defined f(x)=cos(2x+1) invertible? Give reasons. Can you modify domain and co-domain of f such that f becomes invertible?

Show that the exponential function f: R->R , given by f(x)=e^x , is one-one but not onto. What happens if the co-domain is replaced by R0+ (set of all positive real numbers).

Define a modulus function.If the function f:R rarr R is defined by f(x) =|x| ,draw the graph of the function .Write the domain and range of f.(R is the set of real numbers).

Let f={(x,x^2/(1+x^2)),x in R} be a real function from R to R. Determine the domain and range of f.

If f:R rarr R and f(x)=g(x)+h(x) where g(x) is a polynominal and h(x) is a continuous and differentiable bounded function on both sides, then f(x) is one-one, we need to differentiate f(x). If f'(x) changes sign in domain of f, then f, if many-one else one-one. If f:R rarr R and f(x)=2ax +sin2x, then the set of values of a for which f(x) is one-one and onto is

If f:R rarr R and f(x)=g(x)+h(x) where g(x) is a polynominal and h(x) is a continuous and differentiable bounded function on both sides, then f(x) is one-one, we need to differentiate f(x). If f'(x) changes sign in domain of f, then f, if many-one else one-one. If f:R rarr R and f(x)=2ax +sin2x, then the set of values of a for which f(x) is one-one and onto is