fx=3x+2 -1 gx=x-3^-12 +5 fgx=-1013

To find f(g(x)), we substitute g(x) into f(x):

f(g(x)) = 3/(g(x) + 2) - 1

Substituting g(x) = (x-3)^(-1/2) + 5:

f(g(x)) = 3/((x-3)^(-1/2) + 5 + 2) - 1

Simplifying the denominator:

f(g(x)) = 3/((x-3)^(-1/2) + 7) - 1

To find the value of x when f(g(x)) = -10/13, we set the expression equal to -10/13 and solve for x:

-10/13 = 3/((x-3)^(-1/2) + 7) - 1

Multiplying both sides by ((x-3)^(-1/2) + 7):

-10/13 * ((x-3)^(-1/2) + 7) = 3 - ((x-3)^(-1/2) + 7)

Expanding:

-10/13 * (x-3)^(-1/2) - 70/13 = 3 - (x-3)^(-1/2) - 7

Simplifying:

-10/13 * (x-3)^(-1/2) - 70/13 = - (x-3)^(-1/2) - 4

Combining like terms:

-10/13 * (x-3)^(-1/2) + (x-3)^(-1/2) = 70/13 - 4

To simplify the equation further, we can multiply all terms by 13 to eliminate the denominators:

-10 * (x-3)^(-1/2) + (x-3)^(-1/2) = 70 - 52

Simplifying:

-10 * (x-3)^(-1/2) + (x-3)^(-1/2) = 18

Combining like terms:

-9 * (x-3)^(-1/2) = 18

Multiplying both sides by -(1/9):

(x-3)^(-1/2) = -2

To eliminate the exponent, we square both sides:

[(x-3)^(-1/2)]^2 = (-2)^2

Simplifying:

x-3 = 4

Adding 3 to both sides:

x = 7

Therefore, the value of x when f(g(x)) = -10/13 is x = 7