Given f(x)=2-3x and g(x)=hx+k where h and k are constants.If gf(-1)=6 and g^-1(4)=g(4),find the possible values of h and k?
First write expressions for gf(x) and g^-1(x) in terms of h and k.
Then write equations for gf(-1) and g^(-1)(4)
This will leave you with two equations that involve only h and k.
Solve them simultaneously.
For starters,
gf(x) = h(2-3x) + k = 2h -3hx + k
gf(-1) = 2h + 3h + k = 5h + k
Now compute the inverse function of g and complete the solution.
See if you get
g^-1(x) = [(x-k)/h] + k
To solve this problem, we'll start by finding the compositions of the given functions.
1. The composition gf(-1) means "g of f of -1." We substitute -1 into the function f(x):
f(-1) = 2 - 3(-1)
= 2 + 3
= 5
Next, we substitute this result into the function g(x):
g(f(-1)) = g(5) = h(5) + k
Given that g(f(-1)) = 6, we can write:
h(5) + k = 6 (Equation 1)
2. The composition g^-1(4) means "the inverse of g of 4." We substitute 4 into the function g(x):
g(4) = h(4) + k
Given that g^-1(4) = g(4), this means the inverse of g(4) is also equal to g(4). Therefore:
h(4) + k = h^-1(4) + k
To solve for the possible values of h and k, let's consider both equations together:
h(5) + k = 6 (Equation 1)
h(4) + k = h^-1(4) + k
Since k is common to both equations, we can subtract Equation 2 from Equation 1:
h(5) - h(4) = h^-1(4)
To simplify further, let's assign a variable for h^-1(4) and rewrite the equation:
Let p = h^-1(4)
h(5) - h(4) = p
Now, we have two equations:
1. h(5) - h(4) = p (Equation A)
2. h(5) + k = 6 (Equation B)
We need to find the possible values of h and k.
Since we have two equations with two variables, we can solve them simultaneously. Subtract Equation B from Equation A:
(h(5) - h(4)) - (h(5) + k) = p - 6
By simplifying, we get:
-h(4) - k = p - 6
Now, we can substitute the value of p obtained from Equation B into this equation:
-h(4) - k = (h(5) - h(4)) - 6
Simplify further:
-h(4) - k = h(5) - h(4) - 6
The h(4) terms cancel out:
-h(4) - k = h(5) - 6
Now, let's rearrange the equation to isolate k:
k = -(h(5) - 6) + h(4)
Finally, we can identify the possible values of h and k based on the equation:
For any value of h, k = -(h(5) - 6) + h(4).
Therefore, the possible values of h are any real number, and k is calculated based on the given equation.