Problem 25 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(f(x)=e^{x}\) and \(g(x)=\log _{e} x\) then which of the following is true [MPPET-2008] (a) \(f\\{g(x)\\} \neq g\\{f(x)\\}\) (b) \(f\\{g(x)\\}=g\\{f(x)\\}\) (c) \(f\\{g(x)\\}+g\\{f(x)\\}=0\) (d) \(f\\{g(x)\\}-g\\{f(x)\\}=1\)

Step-by-Step Solution

Answer

(b) \( f\{g(x)\} = g\{f(x)\} \)

Step 1: Understand the Given Functions

We have two functions: \( f(x) = e^x \) and \( g(x) = \log_e x = \ln x \). The function \( f(x) = e^x \) is the exponential function, and \( g(x) = \ln x \) is the natural logarithm function.

Step 2: Compose f and g: f(g(x))

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). Hence, \( f(g(x)) = f(\ln x) = e^{\ln x} \). Since \( e^{\ln x} = x \), we have \( f(g(x)) = x \).

Step 3: Compose g and f: g(f(x))

For \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \). Thus, \( g(f(x)) = g(e^x) = \ln(e^x) \). Using the logarithm property \( \ln(a^b) = b \ln a \), \( \ln(e^x) = x \ln e = x \). Hence, \( g(f(x)) = x \).

Step 4: Compare f(g(x)) and g(f(x))

From the previous steps, we found both \( f(g(x)) = x \) and \( g(f(x)) = x \). Therefore, \( f(g(x)) = g(f(x)) \).

Step 5: Answer the Multiple Choice Question

The correct answer is option (b) \( f\{g(x)\} = g\{f(x)\} \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression that involves a constant base raised to a variable exponent. In the most common exponential function, the base is the Euler's number, denoted as \( e \), which is approximately equal to 2.71828.
The function is written as \( f(x) = e^x \).
This type of function grows rapidly as the value of \( x \) increases. Exponential functions are widely used in various fields, including natural sciences, business, and social sciences, due to their unique properties.
Some key points about exponential functions include:

• The rate of growth (or decay) is proportional to the current value, meaning it gets faster as the value grows.
• The function never touches the x-axis; it only approaches it asymptotically.
• Exponential functions exhibit continuous growth, making them applicable in modeling population growth, radioactive decay, and compound interest calculations.

Natural Logarithm

The natural logarithm, represented as \( \ln x \), is the inverse of the exponential function \( e^x \).
This means that if \( y = \ln x \), then \( e^y = x \).
The base of the natural logarithm is \( e \), which differentiates it from common logarithms that use base 10.
Natural logarithms are helpful in solving equations where the exponent contains the variable you need to solve for.
Key characteristics of the natural logarithm include:

• \( \ln e = 1 \) and \( \ln 1 = 0 \), meaning the logarithm of \( e \) itself is one, and the logarithm of 1 is zero.
• \( \ln(ab) = \ln a + \ln b \), displaying a property that logarithms turn products into sums, simplifying many multiplication problems.
• Similar to the exponential function, natural logarithms appear in models like radioactive decay, where they help determine the decay constant or half-life.

Function Operations

Function operations are processes that combine two or more functions through addition, subtraction, multiplication, division, or composition.
Composition of functions in particular is when one function is applied to the result of another function, like \( f(g(x)) \).
This involves substitution of one function's output as the input of another, creating a new composite function.
Understanding function operations is crucial in mathematics, especially when dealing with complex equations and transformations.
Some important points about function operations include:

• In composition, \( f(g(x)) \) means you first apply \( g \) to \( x \), and then apply \( f \) to the result. For example, if \( g(x) = \ln x \) and \( f(x) = e^x \), then \( f(g(x)) = e^{\ln x} = x \).
• The order of composition matters; \( f(g(x)) \) is not necessarily equal to \( g(f(x)) \), although in this specific example they are equal \( (f(g(x)) = g(f(x)) = x) \).
• Function operations also include basic arithmetic combinations, such as \( (f + g)(x) = f(x) + g(x) \) or \( (f \cdot g)(x) = f(x) \cdot g(x) \), allowing for the manipulation and integration of functions for broader application.