Final answer:
The graph of f(3(1/x)) is found by substituting x with 3(1/x) in the original function f(x)=x^3-5, which gives f(3(1/x))=27/x^3-5, corresponding to option D.
Step-by-step explanation:
The student asked what the graph of f(3(1/x)) is if f(x)=x^3-5. To find this, we substitute x in f(x) with 3(1/x), which yields (3/x)^3 - 5. Calculating (3/x)^3, we get 27/x^3. Therefore, f(3(1/x)) = 27/x^3 - 5, which matches option D) f(3(1/x))=27x^3-5 when considering the effects of replacing x with 1/x on the function.
To find the graph of \( f \left( \frac{3}{x} \right) \), we need to substitute \( x \) with \( \frac{3}{x} \) into the original function \( f(x) = x^3 - 5 \). The original function is \( f(x) = x^3 - 5 \). Now let's plug \( \frac{3}{x} \) into the function: \( f \left( \frac{3}{x} \right) = \left( \frac{3}{x} \right)^3 - 5 \) To simplify, we'll cube \( \frac{3}{x} \): \( \left( \frac{3}{x} \right)^3 = \frac{3^3}{x^3} = \frac{27}{x^3} \)
Now, we'll substitute this back into our expression: \( f \left( \frac{3}{x} \right) = \frac{27}{x^3} - 5 \) So, the correct form of \( f \left( \frac{3}{x} \right) \) is: \( f \left( \frac{3}{x} \right) = 27x^{-3} - 5 \)
We can see that the correct answer corresponds to option D: D) \( f \left( \frac{3}{x} \right) = 27x^3 - 5 \) (Note: There is a notation inconsistency in option D, where it should read \( 27x^{-3} - 5 \) to be mathematically correct. However, the intent seems to indicate the transformation \( x \) to \( \frac{3}{x} \), hence the cube of \( \frac{3}{x} \) should indeed be \( \frac{27}{x^3} \) which is \( 27x^{-3} \) when rewritten with a negative exponent.