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A function f: R → R defined by f(x) = 5x4 + 2 is one – one but not onto.
- a)True
- b)False
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A function f: R → R defined by f(x) = 5x4+ 2 is one – one b...
The above statement is false. f is neither one-one nor onto.
For one-one: Consider f(x1) = f(x2)
∴ 5x14 + 2 = 5x24 + 2
⇒ x1 = ± x2.
Hence, the function is not one – one.
For onto: Consider the real number 1 which lies in co-domain R, and let
Clearly, there is no real value of x which lies in the domain R such that f(x) = y.
Therefore, f is not onto as every element lying in the codomain must have a pre-image in the domain.
For one-one: Consider f(x1) = f(x2)
∴ 5x14 + 2 = 5x24 + 2
⇒ x1 = ± x2.
Hence, the function is not one – one.
For onto: Consider the real number 1 which lies in co-domain R, and let
Clearly, there is no real value of x which lies in the domain R such that f(x) = y.
Therefore, f is not onto as every element lying in the codomain must have a pre-image in the domain.
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Community Answer
A function f: R → R defined by f(x) = 5x4+ 2 is one – one b...
Explanation:
One-to-one:
- A function is one-to-one (injective) if each element in the domain maps to a unique element in the codomain.
- To check if a function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at most once, then the function is one-to-one.
- In this case, the function f(x) = 5x^4 + 2 is a polynomial function, and since different x values will result in different y values, the function is one-to-one.
Onto:
- A function is onto (surjective) if every element in the codomain has a preimage in the domain.
- To check if a function is onto, we can see if every element in the codomain is being mapped to by at least one element in the domain.
- In this case, the function f(x) = 5x^4 + 2 is a polynomial function with a leading term of 5x^4. This means that the function is unbounded and does not cover all real numbers. Therefore, the function is not onto.
Therefore, the statement "A function f: R → R defined by f(x) = 5x^4 + 2 is one-one but not onto" is false.
One-to-one:
- A function is one-to-one (injective) if each element in the domain maps to a unique element in the codomain.
- To check if a function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at most once, then the function is one-to-one.
- In this case, the function f(x) = 5x^4 + 2 is a polynomial function, and since different x values will result in different y values, the function is one-to-one.
Onto:
- A function is onto (surjective) if every element in the codomain has a preimage in the domain.
- To check if a function is onto, we can see if every element in the codomain is being mapped to by at least one element in the domain.
- In this case, the function f(x) = 5x^4 + 2 is a polynomial function with a leading term of 5x^4. This means that the function is unbounded and does not cover all real numbers. Therefore, the function is not onto.
Therefore, the statement "A function f: R → R defined by f(x) = 5x^4 + 2 is one-one but not onto" is false.
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A function f: R → R defined by f(x) = 5x4+ 2 is one – one but not onto.a)Trueb)FalseCorrect answer is option 'B'. Can you explain this answer?
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