Consider the following statements: Statement 1: The function f : R → R such that f(x) = x 3for all x ∈ R is one-one Statement 2: f(a) = f(b) ⇒ a = b for all a, b ∈ R if the function f is one-one. Which one of the following is correct in respect of the above statements?
Both the statements are true and Statement 2 is the correct explanation of Statement 1.
Analysis of Function Statements: One-One Property
Let's analyze the given statements regarding functions and the one-one property.
Statement 1: Function f(x) = x³ is One-One
Statement 1 says that the function \(f : R \to R\) defined by \(f(x) = x^3\) for all \(x \in R\) is a one-one function.
To check if a function is one-one (also called injective), we need to see if distinct elements in the domain map to distinct elements in the codomain. Equivalently, a function \(f\) is one-one if for any \(a, b\) in the domain, \(f(a) = f(b)\) implies \(a = b\).
For \(f(x) = x^3\), let's assume \(f(a) = f(b)\) for some \(a, b \in R\). This means:
\(a^3 = b^3\)
Taking the cube root of both sides (which is a well-defined operation for real numbers), we get:
\(\sqrt[3]{a^3} = \sqrt[3]{b^3}\)
\(a = b\)
Since \(f(a) = f(b)\) implies \(a = b\) for all \(a, b \in R\), the function \(f(x) = x^3\) is indeed one-one.
Therefore, Statement 1 is true.
Statement 2: Definition of a One-One Function
Statement 2 gives the condition for a function \(f\) to be one-one. It states that \(f(a) = f(b) \implies a = b\) for all \(a, b \in R\) if the function \(f\) is one-one.
This is precisely the standard definition of a one-one (or injective) function. A function \(f\) from set A to set B is defined as one-one if for any two elements \(a, b \in A\), whenever \(f(a) = f(b)\), it must follow that \(a = b\). The statement accurately reflects this definition for functions on the set of real numbers R.
Therefore, Statement 2 is true.
Relationship Between Statements 1 and 2
Statement 1 claims that \(f(x) = x^3\) is one-one. Statement 2 provides the definition of a one-one function. To verify Statement 1, we used the condition given in Statement 2 (\(f(a) = f(b) \implies a = b\)). The fact that \(a^3 = b^3\) implies \(a = b\) for real numbers confirms that \(f(x) = x^3\) satisfies the condition stated in Statement 2, which is the definition of being one-one.
Hence, Statement 2 provides the exact criterion used to determine and explain why Statement 1 is true. Statement 2 is the definition upon which the truth of Statement 1 is based.
Therefore, Statement 2 is the correct explanation of Statement 1.
Conclusion
Both Statement 1 and Statement 2 are true, and Statement 2 correctly explains why Statement 1 is true by providing the definition of the property mentioned in Statement 1.
Similar Questions
Consider the following statements:
1. The relation f defined by \(f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}\) is a function.
2. The relation g defined by \(g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}\) is a function.
Which of the statements given above is/are correct?
A function satisfies \(f(x-y)=\frac{f(x)}{f(y)}\), where f(y) ≠ 0. If f(1) = 0.5, then what is f(2) + f(3) + f(4) + f(5) + f(6) equal to ?
If \({\rm{f}}\left( {\rm{x}} \right) = \frac{{\rm{x}}}{{{\rm{x}} - 1}},\) then what is \(\frac{{{\rm{f}}\left( {\rm{a}} \right)}}{{{\rm{f}}\left( {{\rm{a}} + 1} \right)}}\) equal to?
Let A = {7, 8, 9, 10, 11, 12, 13; 14, 15, 16} and let f ∶ A → N be defined by f(x) = the highest prime factor of x.
How many elements are there in the range of f?
Let R be a relation from N to N defined by R = {(x, y): x, y ∈ N and x 2 = y 3}. Which of the following are not correct?
1. (x, x) ∈ R for all x ∈ N
2. (x, y) ∈ R ⇒ (y, x) ∈ R
3. (x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R
Select the correct answer using the code given below :
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What is \(\displaystyle\sum_{x=1}^{6} 2^{x}\,f(x)\) equal to?
If f(α) = \(\sqrt{\sec^2\alpha−1}\) , then what is \(\frac{f(\alpha)+f(\beta)}{1−f(\alpha) f(\beta)}\) equal to ?
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Important Questions from Relations and Functions
If the function of \(f(x)=\dfrac{x}{x-1}\) express f(3x) in terms of f(x)
If \(f(x)-\dfrac{1}{1+2^{1/x}}\) then at x = 0 the function is:
If f(x) is a periodic function and a is a positive real number such that f(x + 2α) + f(x) = 0 for all x ∈ ℝ, then the period of f(x) is:
Consider the following statements:
1. The relation f defined by \(f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}\) is a function.
2. The relation g defined by \(g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}\) is a function.
Which of the statements given above is/are correct?
If \(f(x)=\frac{1}{1+x}\), g(x) = f{f(x)} and h(x) = f[f{f(x)}], then the value of f(x).g(x).h(x) is:
