Problem 3 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(e^{x}+e^{f(x)}=e\), then range of the function \(f\) is (A) \((-\infty, 1]\) (B) \((-\infty, 1)\) (C) \((1, \infty)\) (D) \([1, \infty)\)

Step-by-Step Solution

Answer

The range of the function \( f \) is \((- fty, 1]\) which corresponds to option (A).

Step 1: Analyze the given equation

The problem gives us the equation \( e^x + e^{f(x)} = e \). We know \( e = e^1 \), so we can rewrite the equation as \( e^x + e^{f(x)} = e^1 \). This equation states that the sum of two exponential terms is a constant.

Step 2: Re-express the equation

Rewriting, we have \( e^{f(x)} = e - e^x \). We need to find constraints for \( f(x) \) by considering the valid range for \( e^{f(x)} \), which is always positive.

Step 3: Determine the range of \( x \)

Since the exponential function \( e^x \) ranges from \( 0 \) to \( \infty \), for the equation \( e^{f(x)} = e - e^x \) to hold, \( e - e^x \) must be positive. This means \( e^x < e \), so \( x < 1 \).

Step 4: Solve for the range of \( f(x) \)

Substitute \( x < 1 \) into the equation \( e^{f(x)} = e - e^x \). As \( x \) approaches \( 1 \), \( e^x \) approaches \( e \), and \( e^{f(x)} \) approaches \( 0 \), implying that \( f(x) \to -\infty \). Conversely, as \( x \to -\infty \), \( e^x \to 0 \), and \( e^{f(x)} = e - 0 = e \), so \( f(x) = 1 \).

Step 5: Conclusion on the range of \( f \)

It is determined that \( f(x) \) can approach values less than 1 infinitely but never exceed 1. Therefore, the range of \( f \) is \((-fty, 1]\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are represented in the form \(y = a^x\), where \(a\) is a constant and \(x\) is the variable. The base \(a\) is typically a positive real number, and in many cases, it is Euler's number \(e\), approximately 2.71828.
Exponential functions are pivotal in calculus due to their unique property of having a constant rate of growth. Unlike linear functions, whose rate is constant over time, exponential functions grow exponentially faster as the value of \(x\) increases.
Some key properties include:

• Exponential Growth: When \(a > 1\), \(y = a^x\) describes rapid increase as \(x\) increases.
• Exponential Decay: If \(0 < a < 1\), the function describes a decay.
• Always Positive: Exponential functions never become zero or negative.

These characteristics allow exponential functions to model various real-world scenarios, such as population growth, radioactive decay, and even finance-related calculations.

Range of a Function

The range of a function is the set of all possible output values, typically represented on the \(y\)-axis, after substituting the domain (input values) into the function. Knowing the range helps us understand the extremes and limitations of a function's behavior.
Consider the function \(f(x) = e^{f(x)} = e - e^x\) from our problem which illustrates how understanding the inputs (domain) clarifies the range. Specifically, the domain here affects how \(e^x\), an exponential part, behaves:

• \(0 < e^x < e\), implies \(x < 1\).
• This limits \(e^{f(x)}\) to a value between zero and \(e\), further constraining \(f(x)\) to always be less than or equal to 1.

The process of defining this involves substituting boundary values from the domain into the function and observing how outputs change. This gives the range \((-\infty, 1]\) in the exercise.

Function Analysis

Function analysis is the study of the properties and behaviors of mathematical functions. It involves looking at specific characteristics such as domain, range, continuity, and differentiability. Through this analysis, we gain a comprehensive view of how a function behaves and changes across its domain.
One important part of function analysis involves understanding the role of constraints and limits. In our exercise, analyzing the equation \(e^{x} + e^{f(x)} = e\) required setting logical boundaries:

• Investigating conditions like \(e^{f(x)} = e - e^x\) established that expressions must remain positive and less than \(e\).
• This approach limited the domain to \(x < 1\) and consequently determined the range of \(f(x)\) to \((-\infty, 1]\).

Using these tools ensures clarity of how constraints affect functional outputs. It allows us to deduce properties such as the direction or trend the function follows across various intervals.