Question

Let f:R→R and g:R→R be two functions defined by f(x)=logₑ(x²+1)−e⁻ˣ+1 and g(x)=1−2e²ˣ/eˣ. Then, for which of the following range of α, the inequality f(g((α−1)²/3))>f(g(α−5/3)) holds?

A. (−2,−1)

B. (2,3)

C. (−1,1)

D. (1,2)

Answer 1

The question requires comparing values of a composition of two functions, f(g(x)), using properties of logarithms and exponents. However, without specific information or context for simplification, a deterministic answer for the inequality f(g((α-1)²/3)) > f(g(α-5/3)) cannot be provided.

Step-by-step explanation:

To solve the inequality f(g((α-1)²/3)) > f(g(α-5/3)), we must examine the inner functions and apply any properties of logarithms and exponents to simplify. For the functions given, f(x) = loge(x²+1) - e^-x + 1 and g(x) = 1-2e²ʸ/eʸ, we note that the property of logarithms assuming that log(a/b) = log a - log b is relevant. When examining the options provided, we'd notice that we need to find a range for α where the inequality holds true.

Without a specific simplification or evaluation given in the details, we cannot provide a direct answer to the inequality. Normally, we would identify the behavior of f(x) and g(x), apply the composite function f(g(x)), and then compare the outputs for the given α values. However, using the provided information, one would need to manipulate the inequality while respecting the properties of the involved exponential and logarithmic functions.