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    Question

    Which of the following functions is not one-one?

    A f(x) = eˣ Correct Answer Incorrect Answer
    B f(x) = tan x, domain: (-π/2, π/2) Correct Answer Incorrect Answer
    C f(x) = x²+3x+1 Correct Answer Incorrect Answer
    D f(x) = ln x Correct Answer Incorrect Answer

    Solution

    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. f(x) = ex

    • This is an exponential function.
    • It is strictly increasing for all real x, meaning as x increases, f(x) also increases without turning back.
    • Therefore, each input x has a unique output f(x) .
    • So, this function is one-one .
    f(x) = tan x, domain (–π/2, π/2)
    • The tangent function normally repeats values (it's periodic), but in the restricted domain (–π/2 to π/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.
    f(x) = x² + 3x + 1
    • This is a quadratic function , which graphs as a parabola .
    • All quadratics of the form ax² + bx + c (with a ≠ 0) are not one-one over the entire real line because they have a turning point (vertex) .
    • For example:
      f(0) = 0² + 3×0 + 1 = 1
      f(–3) = (–3)² + 3×(–3) + 1 = 9 – 9 + 1 = 1
      So, two different inputs give the same output.
    • Hence, this function fails the one-one condition.
    f(x) = ln x
    • The logarithmic function is strictly increasing for all x > 0.
    • That means if x₁ < x₂, then ln(x₁) < ln(x₂).
    • So, no two different inputs will ever give the same output.
    • Hence, this function is one-one .
    Option (C) — f(x) = x² + 3x + 1 is not one-one because it repeats output values for different inputs due to its parabolic nature.

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