7.
What is the output range of the function 𝑦 = 1 − exp (𝑥) for input values in the interval
−∞ < 𝑥 < ∞?
a. −∞ < 𝑦 < ∞
b. 0 < 𝑦 < ∞
c. −∞ < 𝑦 < 1
d. 0 < 𝑦 < 1

Output Range of y = 1 – exp(x) for All Real x: CSIR NET Solved

The correct answer is c. −∞ < 𝑦 < 1. For the function 𝑦 = 1 − exp(𝑥) over −∞ < 𝑥 < ∞, the output approaches 1 as 𝑥 → −∞ but never reaches it, equals 0 at 𝑥 = 0, and decreases without bound as 𝑥 → ∞ .​

Function Behavior

The exponential exp(𝑥) (or e^x) has domain all real numbers and range (0, ∞), always positive and approaching 0 as 𝑥 → −∞ while growing to ∞ as 𝑥 → ∞. Subtracting from 1 flips this: as 𝑥 → −∞, exp(𝑥) → 0 so 𝑦 → 1 (from below); at 𝑥 = 0, exp(0) = 1 so 𝑦 = 0; as 𝑥 → ∞, exp(𝑥) → ∞ so 𝑦 → −∞ . The function is continuous, strictly decreasing (derivative −exp(𝑥) < 0), and hits every value below 1 exactly once.​

Option Analysis

• a. −∞ < 𝑦 < ∞: Incorrect, as 𝑦 never reaches or exceeds 1 (maximum approaches 1 asymptotically).​

• b. 0 < 𝑦 < ∞: Incorrect, since 𝑦 becomes negative for 𝑥 > 0 and never exceeds 1 .

• c. −∞ < 𝑦 < 1: Correct, covering the horizontal asymptote at 𝑦 = 1 and unbounded decrease.​

• d. 0 < 𝑦 < 1: Incorrect, ignores negative values for 𝑥 > 0 .

Graph Insights