Which of the following functions is not one-one?

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) = e^x

• 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 .

• 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.

• 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.

• 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 .