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
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
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
To find the inverse of the function f(x) = 3 + e^(4−x), we can follow these steps:
Step 1: Replace f(x) with y in the equation:
y = 3 + e^(4−x)
Step 2: Swap the x and y variables:
x = 3 + e^(4−y)
Step 3: Solve the equation for y.
To do this, we can first isolate the exponential term:
e^(4−y) = x − 3
Next, take the natural logarithm of both sides to eliminate the exponential:
ln(e^(4−y)) = ln(x − 3)
Simplifying the left-hand side:
4−y = ln(x − 3)
Finally, solve for y by subtracting 4 from both sides:
y = 4 − ln(x − 3)
So, the inverse of the function f(x) = 3 + e^(4−x) is f^(-1)(x) = 4 − ln(x − 3).
Now, let's determine the domain, range, and asymptotes for both functions.
Domain:
For the original function f(x) = 3 + e^(4−x), the exponential term e^(4−x) is defined for all real numbers. Therefore, the domain of f(x) is (-∞,∞).
For the inverse function f^(-1)(x) = 4 − ln(x - 3), the natural logarithm is defined only for positive values. Hence, the domain of f^(-1)(x) is (3,∞).
Range:
The exponential term e^(4−x) is always positive and greater than zero. Therefore, the range of f(x) is (3,∞).
The range of the inverse function f^(-1)(x) = 4 − ln(x - 3) is (-∞,∞).
Asymptotes:
The original function f(x) = 3 + e^(4−x) does not have any vertical asymptotes.
The inverse function f^(-1)(x) = 4 - ln(x - 3) has a vertical asymptote at x = 3 since the natural logarithm is undefined for x = 3.
To graph both functions on the same coordinate plane, you can plot points for both functions using a table of values or use a graphing calculator or software.
To find the inverse of the function f(x) = 3 + e^(4−x), we can follow these steps:
Step 1: Replace f(x) with y:
y = 3 + e^(4−x)
Step 2: Swap the roles of x and y:
x = 3 + e^(4−y)
Step 3: Solve for y:
x - 3 = e^(4−y)
To isolate e^(4-y), we subtract 3 from both sides:
x - 3 - 3 = e^(4−y) - 3
Simplifying further:
x - 6 = e^(4−y)
Step 4: Take the natural logarithm of both sides:
ln(x - 6) = ln(e^(4−y))
Since ln(e^a) = a, the equation becomes:
ln(x - 6) = 4 - y
Step 5: Solve for y:
y = 4 - ln(x - 6)
So, the inverse of the function f(x) = 3 + e^(4−x) is given by g(x) = 4 - ln(x - 6).
Now, let's find the domain and range of both f(x) and g(x) and their asymptotes:
For the function f(x) = 3 + e^(4−x):
- The domain is all real numbers, since there are no restrictions on x.
- The range is y > 3, since e^(4−x) is always positive and adds to 3.
- There are no asymptotes for this function.
For the inverse function g(x) = 4 - ln(x - 6):
- The domain is x > 6, as the natural logarithm is undefined for non-positive values.
- The range is all real numbers, since ln(x - 6) can take any real value.
- There is a vertical asymptote at x = 6, as the function becomes undefined when x = 6.
To graph both functions on the same coordinate plane, plot some points for each function by selecting different values for x within their respective domains. Connect the points to create the graphs of f(x) and g(x).