Final answer:
The inverse function of f(x) = e^(2x) is not exactly represented in any of the provided answers. The correct inverse would be f-1(x) = ln(x) / 2 or (1/2) * ln(x), which simplifies the property of logarithms allowing the exponent to come out in front as a multiplier. The correct answer is option D. f⁻¹(x) = e^(2x)
Step-by-step explanation:
To find which function is equivalent to f(x) = e^(2x), we need to determine its inverse function, f-1(x). The inverse function 'undoes' the effect of the original function. For exponentials and logarithms, these two types of functions are inverses of each other. Therefore, taking the natural logarithm of both sides of an exponential function gives us the exponent as the result. Applying this to the function in question:
f(x) = e^(2x) becomes x = ln(e^(2x)).
By using the property that the logarithm of a power (such as e^(2x)) allows us to bring the exponent out in front, we can simplify the equation to:
x = 2 * ln(e^x) which simplifies to x = 2 * x because ln(e^x) = x.
From this, we can see that the inverse function is:
f-1(x) = ln(x) / 2 or alternatively expressed as:
f-1(x) = (1/2) * ln(x)
Comparing this with the given options, none match this form exactly. However, to see which option is closest, we factor out a 2 from the natural logarithm in the option that contains it:
f-1(x) = 2ln(2x) can be rewritten as f-1(x) = ln((2x)^2) due to the property that ln(a^b) = b * ln(a).
This demonstrates that option B is the closest, but it is not exactly correct since the inverse function should not have a 2 inside the logarithm.
Thus, the correct answer (which is not provided in the selections) would be:
f-1(x) = ln(x) / 2 or alternatively expressed as f-1(x) = (1/2) * ln(x).