Question

Find \(\left(f^{-1}\right)(a)\) for the function \(f\) and real number \(a\). $$ f(x)=x^{3}-\frac{4}{x}, \quad x>0 \quad a=6 $$

Short answer

The short answer is \( f^{-1}(6) \approx 1.327 \).

Step-by-step solution

Define the Equation

Start by setting the function \( f(x) \) equal to \( a \), i.e. \( x^{3} - \frac{4}{x} = 6 \). This will later allow us to solve for \( x \).

Eliminate the Fraction

To avoid having a fraction in our equation, each side of the equation can be multiplied by \( x \). This will result in a new equation: \( x^{4} - 4 = 6x \).

Rearrange the Equation

The equation \( x^{4} - 6x - 4 = 0 \) needs to be solved for \( x \). However, this equation does not lend itself to straightforward solving techniques such as factoring or square rooting. Instead, a numerical method such as the use of calculators or software to approximate the root should be considered.

Solve the Equation

Numerically solving the equation \( x^{4} - 6x - 4 = 0 \) with the help of calculators or software will yield the root \( x \approx 1.327 \). Thus, the required value of \( x \), which is equivalent to \( f^{-1}(6) \), is approximately 1.327.