12. Prove that f: R → R, [1, if x> 0 f (x) = 10, if x = 0 | -1, if x <0, The mark function provided by is neither unitary nor covert. NCE

The function {:(f:R to R:f(x) = 1, if x gt 0),(" "= 0, if x =0),(" "=-1, if x lt0 is a):}

The function {:(f:R to R:f(x) = 1, if x gt 0),(" "= 0, if x =0),(" "=-1, if x lt0is a):}

Prove that f: (-1,1) rarr R defined by: f(x) = {((x)/(1+x),-1 lt x lt 0),(0,x= 0),((x)/(1-x),0 lt x lt 1):} is bijective. Prove that f is one -one onto

Prove that f(x)= a^x , 0 < a < 1 , is decreasing in R.

Prove that f(x)= a^x , 0 < a < 1 , is decreasing in R.

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f '(x) ne 0 for all x in R . If |[f(x)" "f'(x)], [f'(x)" "f''(x)]|= 0 , for all x in R , then the value of f(1) lies in the interval:

Deine F(x) as the product of two real functions f_1(x) = x, x in R, and f_2(x) ={ sin (1/x), if x !=0, 0 if x=0 follows : F(x) = { f_1(x).f_2(x) if x !=0, 0, if x = 0. Statement-1 : F(x) is continuous on R. Statement-2 : f_1(x) and f_2(x) are continuous on R.

If f: R to [0, infty) defined by f(x) = 10^(x) then f^(-1)(x) =

Let f: R to R be defined by f(x) =x/(x+1), x ne -1 If a,b in R and ab ne 0, f (a/b) + f(b/a) and a+b ne 0 is equal to

Let f_1:R→R,f_2:[0,∞)→R, f_3:R→R and f_4:R→[0,∞) be defined by f_1(x)={ ∣x∣ if x<0 ; e^x if x≥0 ; f_2(x)=x^2 ; f_3(x)={ sin x if x<0 ; x if x≥0 ; f_4(x)={ f_2(f_1(x)) if x<0 f_2(f_1(x)) if x≥0 ​then f_4 is