Find the inverse of the function.
f(x) = e^-x + 5
f(x) = 10^x + 2
To find the inverse of a function, we need to switch the roles of x and y in the function and solve for y.
For the first function, f(x) = e^(-x) + 5, we can start by writing it in terms of y:
y = e^(-x) + 5
Next, switch the x and y variables:
x = e^(-y) + 5
Now, we can solve for y:
x - 5 = e^(-y)
To isolate the exponential term, we subtract 5 from both sides:
x - 5 - 5 = e^(-y) - 5
Simplifying,
x - 10 = e^(-y)
Next, take the natural logarithm (ln) of both sides to get rid of the exponential:
ln(x - 10) = ln(e^(-y))
The natural logarithm and exponential functions are inverses of each other, so they cancel out:
ln(x - 10) = -y
To solve for y, multiply both sides by -1:
-y = -ln(x - 10)
Finally, multiply both sides by -1 again to get the inverse function:
y = ln(x - 10)
Therefore, the inverse of the function f(x) = e^(-x) + 5 is f^(-1)(x) = ln(x - 10).
For the second function f(x) = 10^x + 2, we can go through the same process:
y = 10^x + 2
Switching the variables:
x = 10^y + 2
Subtracting 2 from both sides:
x - 2 = 10^y
Taking the logarithm base 10 (log) of both sides:
log(x - 2) = log(10^y)
Using the fact that the logarithm and exponential functions are inverses:
log(x - 2) = y
Therefore, the inverse of the function f(x) = 10^x + 2 is f^(-1)(x) = log(x - 2).