Question

Find a formula for the inverse of the function.

26. \(h\left( x \right) = \frac{{6 - 3x}}{{5x + 7}}\)

Short answer

The inverse of the function is \({h^{ - 1}}\left( x \right) = \frac{{6 - 7x}}{{5x + 3}}\).

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 function of \(x\), for that interchange\(x\) and \(y\). The equation becomes \(y = {f^{ - 1}}\left( x \right)\).

Determine the formula for the inverse of the function 

Write the given equation as \(y = \frac{{6 - 3x}}{{5x + 7}}\) and solve the equation for \(x\) as shown below:

\(\begin{aligned}y &= \frac{{6 - 3x}}{{5x + 7}}\\y\left( {5x + 7} \right) &= 6 - 3x\\5xy + 7y &= 6 - 3x\\x\left( {5y + 3} \right) &= 6 - 7y\\x &= \frac{{6 - 7y}}{{5y + 3}}\end{aligned}\)

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

\(y = \frac{{6 - 7x}}{{5x + 3}}\)

Thus, the inverse of the function is \({h^{ - 1}}\left( x \right) = \frac{{6 - 7x}}{{5x + 3}}\).