What is g[f(x)-3x] equal to ?
Consider the following for the next two (02) items that follow : Let f(x) and g(x) be two functions such that g(x)=x-\frac{1}{x} and f \circ g(x)=x^{3}-\frac{1}{x^{3}}.
- A. x^{3}-\frac{1}{x^{3}} ✓
- B. x^{3}+\frac{1}{x^{3}}
- C. x^{2}-\frac{1}{x^{2}}
- D. x^{2}+\frac{1}{x^{2}}
Correct Answer: A. x^{3}-\frac{1}{x^{3}}
Explanation
We are given f(g(x)) = x^3 - \frac{1}{x^3}. Note that x^3 - \frac{1}{x^3} = (x - \frac{1}{x})^3 + 3(x - \frac{1}{x}). Replacing (x - \frac{1}{x}) with t, we get f(t) = t^3 + 3t, which implies f(x) = x^3 + 3x. The expression f(x) - 3x evaluates to x^3. Thus, we need to find g(x^3). Since g(x) = x - \frac{1}{x}, substituting x^3 yields g(x^3) = x^3 - \frac{1}{x^3}.
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