Question

If \( f : \mathbb{R} \rightarrow \mathbb{R} \), \( g : \mathbb{R} \rightarrow \mathbb{R} \) are two functions defined by \( f(x) = 2x - 3 \), \( g(x) = x^3 + 5 \), then find \( (f \circ g)^{-1(x) \).}

• A. \( \left( \dfrac{2x + 3}{2} \right)^{\frac{1}{2}} \)
• B. \( \left( \dfrac{x - 7}{2} \right)^{\frac{1}{3}} \)
• C. \( \left( \dfrac{x - 7}{2} \right)^{\frac{1}{2}} \)
• D. \( \left( \dfrac{x + 7}{2} \right)^{\frac{1}{3}} \)

Hint:

To find an inverse of a composite function, first simplify the composition completely and then interchange variables before solving.

Answer 1

The Correct Option is B

Step 1: Find the composite function \( f(g(x)) \).
\[ g(x) = x^3 + 5 \] \[ f(g(x)) = f(x^3 + 5) = 2(x^3 + 5) - 3 = 2x^3 + 7 \]

Step 2: Let \( y = f(g(x)) \) and solve for \( x \).
\[ y = 2x^3 + 7 \] \[ y - 7 = 2x^3 \] \[ x^3 = \dfrac{y - 7}{2} \] \[ x = \left( \dfrac{y - 7}{2} \right)^{\frac{1}{3}} \]

Step 3: Write the inverse function.
\[ (f \circ g)^{-1}(x) = \left( \dfrac{x - 7}{2} \right)^{\frac{1}{3}} \]

Step 4: Conclusion.
Hence, the required inverse function is \[ \boxed{\left( \dfrac{x - 7}{2} \right)^{\frac{1}{3}}} \]