For the given functions f(x)= ln x , g(x)=x-2 and h(x)=5x^2+17x-10, find  fo g o h and state the exact domain of  f o g o h.

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We have to determine fogoh(x) given that

f(x) = lnx

g(x) = x-2

h(x) = 5x^2+17x-10

fogoh(x) = f(g(h(x)))

=> f(g(5x^2+17x-10))

=> f(5x^2+17x-10 - 2)

=> f(5x^2 + 17x - 12)

=> f(5x^2 + 20x - 3x - 12)

=> f(5x(x + 4) - 3(x + 4))

=> f((5x - 3)(x + 4))

=> ln((5x - 3)(x + 4))

=> ln (5x - 3) + ln(x + 4)

The value of fogoh(x) = ln (5x - 3) + ln(x + 4)

The log of negative numbers or 0 is not defined. This gives 5x - 3 > 0 and x + 4 >0

=> x > 3/5 and x > -4

The domain of fogoh(x) is (3/5, inf.)

The required function is fogoh(x) = ln (5x - 3) + ln(x + 4) and the domain of the function is (3/5, inf.)

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