Final answer:
The value of f(g(x)) when f(x) = 15x^3 and g(x)=(5x)^13 are inverses is simply x. This is because applying f on g(x) effectively 'undoes' the transformation of g, leaving the original input x intact.
Step-by-step explanation:
The student's question is about calculating the value of f(g(x)) when f(x) = 15x3 and g(x) = (5x)13 are inverse functions. To find the value of f(g(x)), we substitute g(x) into f(x). Because these two functions are inverses, f(g(x)) should simplify to x.
Let's substitute g(x) into f(x):
f(g(x)) = f((5x)13) = 15 * ((5x)13)3
Since the exponents multiply when we raise a power to a power, we get:
f(g(x)) = 15 * (513)3 * x13*3
Notice that 513 raised to the 3rd power is 539 and 13*3 is 39, which is the reciprocal of 3/39 when we take (15x3)1/39. Therefore:
f(g(x)) = 15 * (539)1/39 * x39/39
This simplifies to:
f(g(x)) = 15/15 * x1 = x
So, the value of f(g(x)) is x.