Question

this function is what onto , many one , one one , into ?

Answer 1

f( x ) = ln{ √(1+x⁴ ) - x²}

first we check is this function one -one ??

we know , that any function y= f(x) will be one-one only when , dy/dx is strictly increasing or strictly decreasing .

now, differentiate f(x) wrt x

dy/dx = 1/{√(1+ x⁴) - x²} × { 4x³/2√(1+x⁴) -2x }

= 1/{ √( 1+x⁴)-x²} ×2{ x³- x√(1+x⁴)}/√(1+x⁴)

= -2×x/√(1+x⁴ )
= -2x/√(1+x⁴)

we know, exponential function always positive so, √(1+x⁴)> 0
but - ∞ < x < ∞
so, dy/dx is positive as well as negative e.g dy/dx >0 or < 0


so , function is many- one function .

Answer 2

we can see that here ...

range of function = co - domain

so function is onto.

but function not strictly increasing and decreasing so it is many -one.

=) function is many -one onto

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hope it will help u