If g(x) = 5 + x + e^x, find g^-1(6)
g^-1(6) = ?
if g^-1 (6) = a
then f(a) = 6
5 + a + e^a = 6
e^a = 1-a
clearly a = 0 is a solution
g^-1 (6) = 0
To find g^(-1)(6), we need to first express g(x) in terms of x explicitly, and then interchange x and g(x) to find the inverse function.
Given g(x) = 5 + x + e^x, we want to solve for x in terms of g(x).
To do this, we can start by subtracting 5 from both sides of the equation:
g(x) - 5 = x + e^x
Next, we can rearrange the equation to isolate the exponential term:
e^x = g(x) - 5 - x
To solve for x, we need to take the natural logarithm (ln) of both sides:
ln(e^x) = ln(g(x) - 5 - x)
This simplifies to:
x = ln(g(x) - 5 - x)
Now that we have x expressed in terms of g(x), we can interchange x and g(x) to find the inverse function:
g^(-1)(x) = ln(x - 5 - g(x))
To find g^(-1)(6), we substitute 6 into the inverse function:
g^(-1)(6) = ln(6 - 5 - g(6))
To figure out g(6), we need to evaluate g(x) by plugging x = 6 into the original function:
g(6) = 5 + 6 + e^6
Now we can substitute g(6) back into the inverse function:
g^(-1)(6) = ln(6 - 5 - (5 + 6 + e^6))
Simplifying further:
g^(-1)(6) = ln(6 - 5 - 11 - e^6)
Finally, we can compute the value of g^(-1)(6) by evaluating the expression inside the natural logarithm.