Question

7-14 determine whether the equation or table defines \(y\) as a function of \(x\).

11. \({\left( {y + 3} \right)^3} + 1 = 2x\)

Short answer

The equation defines \(y\) as a function of \(x\).

Step-by-step solution

State the rule that defines the function

Not every equation defines a function. The equation \({\mathop{\rm y}\nolimits} = {{\mathop{\rm x}\nolimits} ^2}\) defines \(y\) as a function of \(x\) because it determines exactly one valueof \(y\) for each value of\(x\).

Determine whether the equation defines \({\mathop{\rm y}\nolimits} \) as a function of \({\mathop{\rm x}\nolimits} \)

Solve the equation for \({\mathop{\rm y}\nolimits} \) as shown below:

\({\left( {y + 3} \right)^3} + 1 = 2x\)

\({\left( {y + 3} \right)^3} = 2x - 1\)

\(y + 3 = \sqrt[3]{{2x - 1}}\)

\(y = - 3 + \sqrt[3]{{2x - 1}}\)

The equation \({\left( {y + 3} \right)^3} + 1 = 2x\) defines \(y\) as a function of \(x\) because the equation determines exactly one value of \(y\) for each value of \(x\).

Thus, the equation defines \(y\) as a function of \(x\).