Find the inverse of the function below. Graph the function below and the inverse. Determine the domain, range and asymptotes of the function below and the inverse function. Please show all your work.
f(x) = 2e^-x + 5
Just looking at this gives me a headache. HELP!
x = 2 e^-y + 5
2 e^-y = x-5
e^-y = (x-5)/2
ln e^-y= -y = ln [(x-5)/2]
y = - ln [(x-5)/2]
= - [ ln(x-5) - ln 2 ]
y = ln 2 - ln(x-5)
your given function is
y = 2e^-x + 5
to form the inverse, interchange the x and y variables, so the inverse is
x = 2e^-y + 5
the inverse graph will be a reflection of the original graph in the line y = x
Pick a few ordered pairs of the original function, e.g.
(0,7), (1, 5.7) , (3, 5.1) , (-1, 10.4 ) , (-5, large) , (5, just a bit over 5)
sketch the first graph
for the inverse, switch the x and y of the ordered pairs, (same as refection in line y = x)
if you want to express your inverse as a function....
x-5 = 2e^-x
take ln of both sides
ln (x-5) = ln2 + ln e^-y
ln(x-5) - ln2 = -ylne , but lne = 1
y = ln2 - ln(x-5)
or
y = ln(2/(x-5) )
Sure! Let's break it down step by step:
1. Finding the inverse of the function:
To find the inverse of a function, we need to switch the roles of x and y in the original function and solve for y. Let's name the inverse function as g(x).
So, we can rewrite the function as:
x = 2e^(-g(x)) + 5
Now, let's solve for g(x):
x - 5 = 2e^(-g(x))
(x - 5)/2 = e^(-g(x))
To solve for g(x), we take the natural logarithm of both sides:
ln((x - 5)/2) = -g(x)
Finally, we can find g(x) by multiplying both sides by -1:
g(x) = -ln((x - 5)/2)
So, the inverse of the function f(x) = 2e^(-x) + 5 is g(x) = -ln((x - 5)/2).
2. Graphing the function and its inverse:
To graph the function and its inverse, we'll plot the points and draw the curves.
Start by plotting points for the original function f(x) using different values of x. For example, choose some values for x, calculate the corresponding values of f(x) using the function, and plot those points on a graph.
Next, plot points for the inverse function g(x). Use the same values of x that were used for f(x), but this time, calculate the corresponding values of g(x) using the inverse function, and plot those points on the graph.
Finally, connect the plotted points for each function with a smooth curve to graph the original function and its inverse.
3. Determining the domain, range, and asymptotes:
The domain of the original function f(x) is all real numbers since there are no restrictions on the value of x.
The range of f(x) depends on the value of the exponential term e^(-x). Since it always evaluates to a positive number or zero, adding 5 to it will shift the range upward, making it greater than or equal to 5.
For the inverse function g(x), the domain of g(x) consists of all real numbers except for x = 5 (since dividing by zero is undefined).
The range of g(x) includes all real numbers. As x approaches infinity or negative infinity, the logarithmic term in g(x) approaches negative infinity as well.
The original function f(x) has a horizontal asymptote at y = 5, while the inverse function g(x) has a vertical asymptote at x = 5.
By following these steps, you should be able to find the inverse, graph both the function and its inverse, and determine the domain, range, and asymptotes of each.