Question

Find a formula for the inverse of the function.

25. \(g\left( x \right) = 2 + \sqrt {x + 1} \)

Short answer

The inverse of the function is \({g^{ - 1}}\left( x \right) = {\left( {x - 2} \right)^2} - 1\) with the domain \(x \ge 2\).

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 

It is observed that \(g\left( x \right) = 2 + \sqrt {x + 1} \) with \(y \ge 2\).

Write the given equation as \(y = 2 + \sqrt {x + 1} \) and solve the equation for \(x\) as shown below:

\(\begin{aligned}y &= 2 + \sqrt {x + 1} \\y - 2 &= \sqrt {x + 1} \\{\left( {y - 2} \right)^2} &= x + 1\\x &= {\left( {y - 2} \right)^2} - 1\left( {{\mathop{\rm since}\nolimits} \,\,y \ge 2} \right)\end{aligned}\)

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

\(y = {\left( {x - 2} \right)^2} - 1\)

Thus, the inverse of the function is \({g^{ - 1}}\left( x \right) = {\left( {x - 2} \right)^2} - 1\) with the domain \(x \ge 2\).