We are given fuctions h:x --> px+6 and
inverse function h^-1 : x= 2/3 x +q.
Let us write given function in the form of y first.
h:x --> px+6 can be written as
y= px+6.
Let us find it's inverse now.
In order to find the inverse of a function, we need to switch x's and y's and then solve for x.
So, first switching x and y in y= px+6.
We get
x=py+6.
Subtracting 6 from both sides, we get
x-6=py+6-6.
x-6 = py.
Dividing both sides by p, we get
[tex]\frac{(x-6)}{p}=y[/tex]
Therefore, inverse function is
[tex]y=\frac{(x-6)}{p}[/tex]
Or [tex]h^{-1}=\frac{(x-6)}{p}[/tex]
We are given inverse function
[tex]h^{-1} =\frac{2}{3}x+q[/tex]
Equating both inervse functions, we get
[tex]\frac{2}{3}x+q=\frac{(x-6)}{p}[/tex]
Let us write left side as a common denominator and splitting right side into two fractions, we get
[tex]\frac{2}{3}x+\frac{3q}{3} =\frac{(x)}{p}-\frac{6}{p}[/tex]
Or
[tex]\frac{2}{3}x+\frac{3q}{3} =\frac{1}{p}x-\frac{6}{p}[/tex]
On comparing left side and right side, we get
[tex]\frac{2}{3}=\frac{1}{p} \ and \ \ \ \frac{3q}{3}=\frac{6}{p}[/tex]
On corss multipication [tex]\frac{2}{3}=\frac{1}{p}[/tex] ,we get
2p = 3.
Dividing both sides by 2, we get
p =3/2.
Let us solve [tex]-\frac{3q}{3}=\frac{6}{p}[/tex] now.
On cross multiplication, we get
-3pq = 18
Plugging value of p, we get
-3(3/2)q = 18
-9/2 q = 18.
Multiplying both sides by 2, we get
-9q =36.
Dividing both sides by 9, we get
-q=4.
Therefore, values p=3/2 and q = -4.
