Find the inverse of the function $y=x^2-12$.
The objective is to find the inverse relation for the function \(y = x^2 - 12\). Finding an inverse relation involves interchanging the role of the input \(x\) and the output \(y\) in the original equation and then isolating the new output variable to express it in terms of the new input variable. We note that the graph of \(y = x^2 - 12\) is a parabola, and its inverse relation, obtained by swapping \(x\) and \(y\), results in a sideways parabola. Since a vertical line test would fail for a sideways parabola, the inverse is technically a relation rather than a single-valued function unless the original domain is restricted. However, in algebraic terms, we can find the expression representing this relation.
To find the inverse relation, we begin with the original equation:
\[y = x^2 - 12\]
By swapping the variables \(x\) and \(y\), we obtain the equation that defines the inverse relation:
\[x = y^2 - 12\]
Our goal now is to isolate \(y\) on one side of the equation.
Starting from \(x = y^2 - 12\), we isolate \(y^2\) by adding 12 to both sides of the equation:
\[x + 12 = y^2\]
To solve for \(y\), we take the square root of both sides. When taking the square root of both sides of an equation of the form \(y^2 = a\), we must consider both positive and negative roots:
\[y = \pm \sqrt{x + 12}\]
Thus, the inverse relation is given by \(y = \pm \sqrt{x + 12}\).
Comparing our result \(y = \pm \sqrt{x + 12}\) with the provided choices:
Our derived expression matches choice (b). Therefore, the correct inverse relation is \(y = \pm \sqrt{x + 12}\).
In real-world applications, inverse relations are essential for reversing processes. For example, if you know the cost \(y\) of a project based on the time \(x\) spent on it as \(y = x^2 - 12\), finding the inverse relation allows you to determine the time required based on a target budget. Understanding how to find and manipulate inverse relations is a fundamental skill in algebra, physics, and economics for solving for independent variables when only dependent outcomes are known.