Question

\( f(x) = 2x - 3 \quad \text{and} \quad g(x) = x^3 + 5 \), then find \( [f \circ g]^{-1}(-9) = ? \)

Answer 1

Step 1: Find the composition \( f \circ g(x) \):
We are given the functions: \[ f(x) = 2x - 3 \quad \text{and} \quad g(x) = x^3 + 5 \] To find the composition \( f(g(x)) \), we substitute \( g(x) \) into \( f \): \[ f(g(x)) = 2(g(x)) - 3 = 2(x^3 + 5) - 3 = 2x^3 + 10 - 3 = 2x^3 + 7 \] So, \( f(g(x)) = 2x^3 + 7 \).

Step 2: Find the inverse of \( f(g(x)) \):
Now, we need to find the inverse of \( f(g(x)) = 2x^3 + 7 \). Let \( y = f(g(x)) \), so: \[ y = 2x^3 + 7 \] To find the inverse, we swap \( x \) and \( y \), and solve for \( y \): \[ x = 2y^3 + 7 \] Solving for \( y \): \[ 2y^3 = x - 7 \] \[ y^3 = \frac{x - 7}{2} \] Taking the cube root of both sides: \[ y = \sqrt[3]{\frac{x - 7}{2}} \] So, the inverse function is: \[ [f \circ g]^{-1}(x) = \sqrt[3]{\frac{x - 7}{2}} \]

Step 3: Evaluate the inverse at \( x = -9 \):
Now, we substitute \( x = -9 \) into the inverse function: \[ [f \circ g]^{-1}(-9) = \sqrt[3]{\frac{-9 - 7}{2}} = \sqrt[3]{\frac{-16}{2}} = \sqrt[3]{-8} = -2 \]

Final Answer:
Therefore, \( [f \circ g]^{-1}(-9) = -2 \).