calculus

Find the inverse of the following function. Find the domain, range, and asymptotes of each function. Graph both functions on the same coordinate plane.

f(x)=3+e^4−x

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  1. I think you mean f(x) = 3 + e^(4−x)

    To get the inverse, replace the f(x) by x, and the x by f'(x), then solve for f'(x):
    f(x) = 3 + e^(4−x)
    x = 3 + e^(4 - f'(x))
    x - 3 = e^(4 - f'(x))
    ln(x - 3) = 4 - f'(x)
    f'(x) = 4 - ln(x - 3)
    This is the inverse of the original function.

    i. Domain
    Domain is the set of all possible values of x.
    f(x) = 3 + e^(4−x)
    For this function, the domain is all real numbers.

    Now, try to determine the domain of the inverse function, f'(x) = 4 - ln(x - 3)

    ii. Range
    Range is the set of all possible values of f(x).
    f(x) = 3 + e^(4−x)
    For this function, the range is all real numbers greater than 3. Note that the smallest value that e^(4-x) approaches, is zero, which happens if x is very large.

    Now, try to determine the range of the inverse function, f'(x) = 4 - ln(x - 3)

    iii. Asymptote
    To get the horizontal asymptote, we get the limit of the function as x -> infinity.
    To get the vertical asymptote, we get the limit of the function as f(x) -> infinity.

    f(x) = 3 + e^(4−x)
    Horizontal Asymptote: f(x) = 3
    Vertical Asymptote: none

    Now, try to determine the asymptotes of the inverse function, f'(x) = 4 - ln(x - 3)

    Hope this helps :3

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  2. I will assume you meant
    f(x) = 3 + e^(4-x)
    or else you would just have a straight line

    since e^? can have any real number as an exponent,
    the domain of the function is any real number
    for the range ...
    as x ---> positive large, e^(4-x) becomes very small
    e.g. e^-100 = 3.7x10^-44
    the function approaches 3 + 0 which is 3
    as x ---> negative large, e^(4-x) approaches infinitiy.
    so for the original f(x)
    domain: any real number
    range: any real number, y>3

    inverse:
    step 1: for y = 3 + e^(4-x), interchange the x and y variables to get
    x = 3 + e^(4-y)
    x-3 = e^(4-y)
    take ln of both sides
    ln(x-3) = ln e^(4-y)
    ln(x-3) = 4-y

    y = 4 - ln(x-3)

    of course the domain of the original becomes the range of the inverse
    and the range of the original becomes the domain of the inverse

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