Question at position 2 Which of the following is true? If f(x)=exf\left(x\right)=e^x, then:The domain of f is all real numbers except zero and the range is all real numbers.Both the domain and range of f are all real numbers.Both the domain and range of f are all positive real numbers.The domain of f is all real numbers and the range is all positive real numbers.The domain of f is all positive real numbers and the range is all real numbers.

Question

Question at position 2 Which of the following is true? If f(x)=exf\left(x\right)=e^x, then:The domain of f is all real numbers except zero and the range is all real numbers.Both the domain and range of f are all real numbers.Both the domain and range of f are all positive real numbers.The domain of f is all real numbers and the range is all positive real numbers.The domain of f is all positive real numbers and the range is all real numbers.

BlackTom AI · Accepted Answer

The problem asks us to evaluate the domain and range of f(x) = e^x based on the given statements.
Option 1: 'The domain of f is all real numbers except zero and the range is all real numbers.' This is incorrect because the exponential function e^x is defined for every real x, so its domain is all real numbers, and its range is strictly positive, not all real numbers.
Option 2: 'Both the domain and range of f are all real numbers.' While the domain is indeed all real numbers, the range is not; e^x never equals zero or a negative number, so the range is not all real numbers.
Option 3: 'Both the domain and range of f are all positive real numbers.' The domain includes negative x values as well, so it is not limited to positive real numbers; thus this statement is false. Moreover, the range is positive but the domain is not restricted to positive values.
Option 4: 'The domain of f is all real numbers and the range is all positive real numbers.' This matches the standard properties of the exponential function: e^x is defined for every real x, and e^x > 0 for all x, giving a range of (0, ∞).
Option 5: 'The domain of f is all positive real numbers and the range is all real numbers.' This reverses the actual domains and ranges and is incorrect for both parts; the domain is not restricted to positive numbers and the range is not all real numbers.

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