Answer-Which of the following functions is not one-one?

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A one-one function injective is a function where no two different inputs give the same output. Graphically, this means the function must pass the horizontal line test  any horizontal line should intersect the graph at most once.   fx  ex   This is an exponential function.  It is strictly increasing for all real x, meaning as x increases, fx also increases without turning back.  Therefore, each input x has a unique output fx .  So, this function is one-one .    fx  tan x, domain 2, 2   The tangent function normally repeats values its periodic, but in the restricted domain 2 to 2, it is strictly increasing and continuous.  This domain avoids the asymptotes and keeps the function monotonic.  Within this interval, every x gives a unique output , and no repetition occurs.  So, this is one-one in the given domain.    fx  x  3x  1   This is a quadratic function , which graphs as a parabola .  All quadratics of the form ax  bx  c with a  0 are not one-one over the entire real line because they have a turning point vertex .  For example f0  0  30  1  1 f3  3  33  1  9  9  1  1 So, two different inputs give the same output.  Hence, this function fails the one-one condition.    fx  ln x   The logarithmic function is strictly increasing for all x  0.  That means if x  x, then lnx  lnx.  So, no two different inputs will ever give the same output.  Hence, this function is one-one .   Option C  fx  x  3x  1 is not one-one because it repeats output values for different inputs due to its parabolic nature.

Which of the following functions is not one-one?

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