Problem 86 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)\) is defined on \((0,1)\), then the domain of definition of \(f\left(e^{x}\right)+f(\log |x|)\) is (A) \((-e,-1)\) (B) \((-e,-1) \cup(1, e)\) (C) \((-\infty,-1)\) (D) \((-e, e)\)

Step-by-Step Solution

Answer

The answer is (B) \((-e, -1) \cup (1, e)\).

Step 1: Understanding the Problem

We need to find the domain of the expression \(f(e^x) + f(\log |x|)\) given that \(f(x)\) is defined on \((0,1)\). This means \(f(x)\) is only valid when the input \(x\) is within the interval \((0,1)\).

Step 2: Analyzing \(f(e^x)\)

For \(f(e^x)\) to be defined, \(e^x\) must lie in the interval \((0,1)\). Since \(e^x > 0\) for all real \(x\), \(e^x < 1\) must be true. This inequality simplifies to \(x < 0\). So, the condition for \(f(e^x)\) is \(x \in (-\infty, 0)\).

Step 3: Analyzing \(f(\log |x|)\)

For \(f(\log |x|)\) to be defined, \(|x|\) must be in the interval \((0,1)\), which implies \(0 < |x| < 1\). Therefore, \(x\) must be in the union of intervals \((-1,0)\) and \((0,1)\). Then \(\log |x|\) would indeed be between 0 and 1.

Step 4: Combining Intervals for Valid Domain

We have two conditions: \(x \in (-\infty, 0)\) from \(f(e^x)\) and \(x \in (-1, 0) \cup (0, 1)\) from \(f(\log |x|)\). The intersection of these two sets is \(x \in (-1, 0)\). Therefore, for both parts to be defined, \(x\) must be in the interval \((-1, 0)\).

Step 5: Examining the Options

Now that we know \(x \in (-1, 0)\), we compare this with the given multiple-choice options. The option \((-e, -1) \cup (1, e)\) includes \((-1, 0)\) within \((-e, -1)\). Thus, the correct choice is (B).

Key Concepts

Interval notation

To express a set of numbers representing part of the domain of a function, we often use interval notation. Interval notation provides a simple way to describe which numbers are included in a set or range. For example:

• The interval \((a, b)\) denotes all numbers between \(a\) and \(b\) but not including \(a\) and \(b\) themselves.
• Similarly, \([a, b]\) means \(a\) and \(b\) are included in the interval.
• If the interval is open on the left and closed on the right, it is written as \((a, b]\). Conversely, \([a, b)\) would signify a closed interval on the left and open on the right.
• Intervals can also include approaches to infinity, like \((-\infty, a)\) or \((b, \infty)\).

Understanding interval notation is crucial for precisely conveying the domain and range of functions. In the given problem, the solution reveals the domain through intervals; specifically, finding an intersection of intervals like \((-1, 0)\). This means that the valid inputs for the given function fall within this narrow range of numbers.

Logarithmic function

Logarithmic functions are an essential concept in mathematics, particularly in understanding behavioral patterns of growth and decay. A logarithmic function is generally expressed as \(y = \log_b(x)\), where \(b\) is the base, and it must be a positive number different from 1.

• The logarithm \(\log_b(x)\) is the exponent to which the base \(b\) must be raised to produce the number \(x\).
• For example, if \(b = 10\), then \(\log_{10}(100) = 2\) because \(10^2 = 100\).
• Logarithms have properties that simplify the manipulation and calculation of large numbers, among them, the product, quotient, and power rules.

In the exercise, \(f(\log |x|)\) requires the input \(\log |x|\) to be within the function's domain, which is \((0,1)\). This ensures that the value inside the logarithm, \(|x|\), is appropriately chosen such that it stays valid within the range set by the constraint.

Exponential function

Exponential functions are characterized by their constant percentage growth or decay, distinct from linear functions. An exponential function has the form \(y = a \, e^{bx}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.718.

• In mathematics, \(e^x\) is one of the most significant exponential functions due to its continual growth pattern. It's distinct because the rate of change of \(e^x\) is proportional to the function itself.
• These functions are used in various fields to model phenomena that grow at a consistent rate, such as populations, finance compounding, or radioactive decay.
• One crucial property is that they never reach zero; \(e^x\) is always positive regardless of the value of \(x\).

For the question at hand, \(f(e^x)\) is valid when \(e^x\) fits within the defined interval \((0,1)\). Solving \(e^x < 1\) helps in pinpointing the conditions under which the function makes sense, which translates to knowing where \(x\) must lie to ensure that \(f(e^x)\) is properly defined.