Question

Find a formula for the inverse of the function.

29. \(y = {\left( {2 + \sqrt(3){x}} \right)^3}\)

Short answer

The inverse function is \(y = {\left( {\sqrt(5){x} - 2} \right)^3}\).

Step-by-step solution

Condition to find the inverse function of a one-to-one function 

Step 1: Write the equation as \(y = f\left( x \right)\).

Step 2: If possible, solve the equation in terms of \(y\).

Step 3: Express the inverse \({f^{ - 1}}\) as a functionof \(x\), for that interchange\(x\) and \(y\). The equation becomes \(y = {f^{ - 1}}\left( x \right)\).

Determine the formula for the inverse of the function

Solve the equation for \(x\) as shown below:

\(\begin{aligned}\sqrt(5){y} & 2 + \sqrt(3){x}\\\sqrt(3){x} &= \sqrt(5){y} - 2\\x &= {\left( {\sqrt(5){y} - 2} \right)^3}\end{aligned}\)

Interchange \(x\) and \(y\) in the above equation as shown below:

\(y = {\left( {\sqrt(5){x} - 2} \right)^3}\)

Thus, the inverse function is \(y = {\left( {\sqrt(5){x} - 2} \right)^3}\).