find the inverse of the following function. Find the domain, range, and asymptotes of each function.
f(x)=3+1n(x+2)
I think you meant to type
f(x) = 3 + ln(x+2)
we can take logs of only positive numbers, so
x+2 > 0
x > -2
ln(a number) can be any real number.
domain: x > 2
range: f(x) any real number
asymptote: none
of course, there is a vertical asymptote at x = -2
To find the inverse of the function f(x) = 3 + ln(x + 2), we need to follow these steps:
Step 1: Replace f(x) with y: y = 3 + ln(x + 2).
Step 2: Swap x and y: x = 3 + ln(y + 2).
Step 3: Solve for y: To isolate y, subtract 3 from both sides: x - 3 = ln(y + 2).
Step 4: Convert the equation to exponential form: Rewrite the equation as e^(x - 3) = y + 2.
Step 5: Subtract 2 from both sides to get y alone: y = e^(x - 3) - 2.
Therefore, the inverse of f(x) is given by g(x) = e^(x - 3) - 2.
Now, let's determine the domain, range, and asymptotes of both f(x) and g(x):
For the original function f(x) = 3 + ln(x + 2):
Domain: The natural logarithm is defined only for x + 2 > 0, so x + 2 > 0 => x > -2. Hence, the domain of f(x) is x > -2.
Range: Since ln(x + 2) can take any value, the range of f(x) is (-∞, +∞).
Asymptotes: The natural logarithm function has a horizontal asymptote at y = 0 (x-axis). So, there is a horizontal asymptote at y = 3.
For the inverse function g(x) = e^(x - 3) - 2:
Domain: The exponential function e^(x - 3) is defined for all real numbers, so the domain of g(x) is (-∞, +∞).
Range: The range of g(x) is (-2, +∞), since e^(x - 3) can never be less than -2.
Asymptotes: The exponential function e^(x - 3) has no asymptotes.
In summary:
Original function f(x): Domain: x > -2, Range: (-∞, +∞), Asymptotes: y = 3.
Inverse function g(x): Domain: (-∞, +∞), Range: (-2, +∞), Asymptotes: None.