Given f(x) = eˣ, which of the following gives the solutions to the inequality f(x) > 0?
A) No real numbers
B) All real numbers greater than or equal to 1
C) All real numbers
D) All positive real numbers
E) All real numbers greater than 1
IMAT 2026 · Question 56
Q56 - Logarithms, composite and inverse functions
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The correct answer is C) All real numbers.
eˣ is positive for every real x. The base e ≈ 2.718 is positive, and any power of a positive number is positive. Negative exponents just give fractions: e⁻² = 1/e² ≈ 0.135, still above 0.
On the graph, eˣ always lies above the x-axis: it gets closer and closer to 0 as x → −∞, but never reaches it.
A) Incorrect: eˣ > 0 is true for every x, not for none.
B) Incorrect: Being greater than or equal to 1 is not required: even negative x gives positive values.
C) Correct: eˣ > 0 for every real x.
D) Incorrect: This confuses eˣ with ln x, which is defined only for positive x.
E) Incorrect: eˣ > 0 holds for every x. (eˣ > 1 holds for x > 0, but that is a different question.)
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