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If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.?
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If R is the set of all real numbers,then the function f: R→R defined b...
Function f: R→R defined by f(x) = 2^x
One-One Function
A function is one-one if each element in the domain maps to a unique element in the range. To determine if the function f(x) = 2^x is one-one, we need to check if different elements in the domain map to different elements in the range.
Onto Function
A function is onto if every element in the range is mapped to by at least one element in the domain. To determine if the function f(x) = 2^x is onto, we need to check if every element in the range has at least one pre-image in the domain.
Analysis
- The function f(x) = 2^x is one-one into, but not onto.
- To prove that the function is one-one, we can use the fact that if f(a) = f(b), then a = b. If we assume that 2^a = 2^b, then we can take the logarithm of both sides to get a = b. Therefore, different elements in the domain map to different elements in the range.
- To prove that the function is not onto, we can observe that there is no real number x such that 2^x = -1. Therefore, the element -1 in the range does not have a pre-image in the domain.
- Hence, the function f(x) = 2^x is one-one into, but not onto.
Conclusion
The function f: R→R defined by f(x) = 2^x is one-one into, but not onto.
One-One Function
A function is one-one if each element in the domain maps to a unique element in the range. To determine if the function f(x) = 2^x is one-one, we need to check if different elements in the domain map to different elements in the range.
Onto Function
A function is onto if every element in the range is mapped to by at least one element in the domain. To determine if the function f(x) = 2^x is onto, we need to check if every element in the range has at least one pre-image in the domain.
Analysis
- The function f(x) = 2^x is one-one into, but not onto.
- To prove that the function is one-one, we can use the fact that if f(a) = f(b), then a = b. If we assume that 2^a = 2^b, then we can take the logarithm of both sides to get a = b. Therefore, different elements in the domain map to different elements in the range.
- To prove that the function is not onto, we can observe that there is no real number x such that 2^x = -1. Therefore, the element -1 in the range does not have a pre-image in the domain.
- Hence, the function f(x) = 2^x is one-one into, but not onto.
Conclusion
The function f: R→R defined by f(x) = 2^x is one-one into, but not onto.
Community Answer
If R is the set of all real numbers,then the function f: R→R defined b...
One- one into
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If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.?
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If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.?.
If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If R is the set of all real numbers,then the function f: R→R defined by f(x) = 2^x is : (a) one-one onto (b) one-one into (c) many-one into (d) many-one onto.?.
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