Problem 23
Question
Let \(f(x)\) be a continuous function defined for \(1 \leq x \leq 3\). If \(f(x)\) takes rational values for all \(x\) and \(f(2)=10\), then \(f(1.5)\) is equal to (A) 0 (B) 10 (C) not defined (D) any constant
Step-by-Step Solution
Verified Answer
10
1Step 1: Understand the Problem
We need to determine the value of the function \( f(x) \) at \( x = 1.5 \). The function \( f(x) \) is defined as continuous and takes rational values for \( 1 \leq x \leq 3 \), with \( f(2) = 10 \).
2Step 2: Analyze Continuity with Rational Values
Continuity implies that there are no jumps or gaps in the values of \( f(x) \) over the interval. Since \( f(x) \) is continuous and takes only rational values on the interval \([1, 3]\), \( f(x) \) must be constant on this interval. Otherwise, there would be a jump between different rational values, contradicting the continuity.
3Step 3: Use Given Point to Determine Constant Value
Since \( f(x) \) is constant due to its rational and continuous nature, \( f(x) \) has the same value for all \( x \) between 1 and 3. Given \( f(2) = 10 \), this constant value must be 10.
4Step 4: Apply Conclusion to Find \( f(1.5) \)
Apply the constancy to find \( f(1.5) \). Since \( f(x) \) is constant on the interval \( [1, 3] \), we have \( f(1.5) = 10 \).
Key Concepts
Rational ValuesConstant FunctionJEE Main Mathematics
Rational Values
Rational values are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. This means any number that can be written as \( \frac{p}{q} \) where \( p \) and \( q \) are integers. Examples of rational values include fractions like \( \frac{1}{2} \), whole numbers like 3 (since it can be written as \( \frac{3}{1} \)), and zero.
In the context of the function \( f(x) \), if \( f(x) \) takes rational values for every \( x \) in the interval \([1, 3]\), then for any point within or on the boundaries of this interval, \( f(x) \) must yield a rational number. This constraint turns out to be quite important when combined with other properties, such as continuity.
For a continuous function that only yields rational numbers in an interval, it would mean it cannot jump between different rational values without becoming discontinuous. Hence, in such contexts, like the exercise here, the function often simplifies to a constant function.
In the context of the function \( f(x) \), if \( f(x) \) takes rational values for every \( x \) in the interval \([1, 3]\), then for any point within or on the boundaries of this interval, \( f(x) \) must yield a rational number. This constraint turns out to be quite important when combined with other properties, such as continuity.
For a continuous function that only yields rational numbers in an interval, it would mean it cannot jump between different rational values without becoming discontinuous. Hence, in such contexts, like the exercise here, the function often simplifies to a constant function.
Constant Function
A constant function is a function that gives the same output value regardless of the input. This is reflected in its graph, which is a horizontal line. For such a function, if \( f(a) = c \), then \( f(x) = c \) for all \( x \) in the domain of the function.
In our particular example with the continuous function \( f(x) \) taking only rational values over an interval, the function must be constant to avoid any discontinuities. It cannot change values from one rational number to another without leaving rationality or introducing a gap, which contradicts the property of continuity.
Thus, since \( f(x) \) is a constant function here and \( f(2) = 10 \), \( f(x) \) must equal 10 for all \( x \) between 1 and 3. This makes solving for \( f(1.5) \) straightforward, since \( f(1.5) \) must also equal 10.
In our particular example with the continuous function \( f(x) \) taking only rational values over an interval, the function must be constant to avoid any discontinuities. It cannot change values from one rational number to another without leaving rationality or introducing a gap, which contradicts the property of continuity.
Thus, since \( f(x) \) is a constant function here and \( f(2) = 10 \), \( f(x) \) must equal 10 for all \( x \) between 1 and 3. This makes solving for \( f(1.5) \) straightforward, since \( f(1.5) \) must also equal 10.
JEE Main Mathematics
JEE Main is an entrance exam in India that is essential for students who wish to pursue undergraduate engineering programs. Mathematics is a significant portion of this examination, covering various complex topics like functions, calculus, algebra, and more.
Problems such as the one we discussed often appear in this exam, testing the students' understanding of continuity, algebraic properties, and the nature of functions. Such exercises require a strong conceptual grasp as students must apply theoretical knowledge to specific and sometimes tricky problems.
To excel in JEE Main Mathematics, students must be adept at recognizing properties of functions, such as when a function might be constant given continuity and rationality. This type of problem-solving is crucial and forms a core part of the Mathematics section of the exam.
Problems such as the one we discussed often appear in this exam, testing the students' understanding of continuity, algebraic properties, and the nature of functions. Such exercises require a strong conceptual grasp as students must apply theoretical knowledge to specific and sometimes tricky problems.
To excel in JEE Main Mathematics, students must be adept at recognizing properties of functions, such as when a function might be constant given continuity and rationality. This type of problem-solving is crucial and forms a core part of the Mathematics section of the exam.
Other exercises in this chapter
Problem 21
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View solution Problem 24
If \(f(x)=\int_{0}^{x} t \cos \frac{1}{t} d t\), then the number of points of discontinuity of \(f(x)\) in the interval \((0, \pi)\) is (A) 1 (B) 2 (C) 0 (D) No
View solution Problem 25
If \(f(x)=(-1)^{\left[x^{2}\right]}\), where \([.]\) denotes the greatest integer function, then (A) \(f(x)\) is discontinuous for \(x=n^{1 / 3}\), where \(n \i
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