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Statement-1 : f(x) = x^(7) + x^(6) - x^(...

Statement-1 : f(x) = x^(7) + x^(6) - x^(5) + 3` is an onto function.
and
Statement -2 : f(x) is a continuous function.

A

Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1.

C

Statement -1 is False, Statement -2 is False

D

Statement -1 is False, Statement -2 is True

Text Solution

AI Generated Solution

The correct Answer is:
To determine the validity of the statements provided about the function \( f(x) = x^7 + x^6 - x^5 + 3 \), we will analyze each statement step by step. ### Step 1: Analyze Statement 1 **Statement 1:** \( f(x) \) is an onto function. 1. **Identify the function type:** The function \( f(x) = x^7 + x^6 - x^5 + 3 \) is a polynomial function of degree 7. 2. **Determine the behavior of odd-degree polynomials:** Odd-degree polynomial functions have the property that their range is all real numbers (\( \mathbb{R} \)). This is because as \( x \) approaches \( +\infty \), \( f(x) \) also approaches \( +\infty \), and as \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \). 3. **Conclusion for Statement 1:** Since the range of \( f(x) \) is \( \mathbb{R} \) and the domain is also \( \mathbb{R} \), \( f(x) \) is indeed an onto function. Therefore, Statement 1 is **true**. ### Step 2: Analyze Statement 2 **Statement 2:** \( f(x) \) is a continuous function. 1. **Identify the type of function:** The function \( f(x) \) is a polynomial function. 2. **Properties of polynomial functions:** All polynomial functions are continuous over their entire domain. This means that there are no breaks, jumps, or asymptotes in the graph of the function. 3. **Conclusion for Statement 2:** Since \( f(x) \) is a polynomial function, it is continuous. Therefore, Statement 2 is also **true**. ### Final Conclusion Both statements are true. Statement 1 is true because \( f(x) \) is an onto function, and Statement 2 is true because \( f(x) \) is continuous. ### Summary of Results - Statement 1: True (since \( f(x) \) is onto) - Statement 2: True (since \( f(x) \) is continuous)
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