What is the inverse function of h(x)= 3-(x+4)/(x-7) and how do you evaluate h^-1(9)?

2 Answers
Jun 11, 2015

h^(-1)(y)=(7y-25)/(y-2) where y!=2.
h^(-1)(9)=38/7

Explanation:

h(x)= 3-(x+4)/(x-7) having x!=7
I find the common denominator and sum all together:
h(x)=(3x-21)/(x-7)-(x+4)/(x-7)
h(x)=(2x-25)/(x-7)
Now I simplify making the division of the 2 polynomials and I obtain:
"quotient"=2, "reminder"=-11
So I can write the function as:
h(x)=2(x-7)/(x-7)-11/(x-7)
h(x)=2-11/(x-7).

Now, the question is how to find the inverse function? Firstly, I try to isolate x:
(x-7)(h(x)-2)=-11
(x-7)=-11/(h(x)-2)
x=-11/(h(x)-2)+7
Therefore we rewrite better the function:
h^(-1)(y)=-11/(y-2)+7=(7y-14-11)/(y-2)=(7y-25)/(y-2).
So we can state that:
h^(-1)(y)=(7y-25)/(y-2) where y!=2.

If we want to find h^(-1)(9):
h^(-1)(9)=(7*9-25)/(9-2)=(63-25)/7=38/7

Jun 11, 2015

The inverse function is h^(-1)(x) = (7x-25)/(x-2).
h^(-1)(x) = 38/7

Explanation:

Since the original function is not that complex, you can determine its inverse function faster by solving the function for x and switching the result for h(x).

h(x) = 3 - (x+4)/(x-7)

h(x) = (3(x-7)- (x+4))/(x-7) = (3x - 21 -x - 4)/(x-7)

h(x) = (2x - 25)/(x-7)

Solve this form of the equation for x to get

h^(x) * (x-7) = 2x-25

x * h(x) - 7 * h(x) * 7 = 2x-25

x * h(x) - 2x = 7 * h(x) - 25

x( h(x) - 2) = 7 * h(x) - 25

x = (7 * h(x) - 25)/(h(x) - 2)

Once you isolate x, simply switch h(x) for x to get the inverse function

h^(-1)(x) = (7x - 25)/(x-2)

To evaluate h^(-1)(9), simply substitute x with 9 to get

h^(-1)(9) = (7 * 9 - 25)/(9-2) = 38/7