Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Show that \(f\) and \(g\) are inverse functions (a) algebraically, (b) graphically, and (c) numerically. $$f(x)=-\sqrt{x-8} ; \quad g(x)=8+x^{2}, \quad x \leq 0$$

Step-by-Step Solution

Answer

The proofs: Algebraic - \(f(g(x)) = -x\) and \(g(f(x)) = x\), Graphical - the graph of \(f\) and \(g\) are reflections of each other across the line \(y = x\), and Numerical - from chosen values, \(f(g(x)) = x\) and \(g(f(x)) = x\) verify that \(f\) and \(g\) are inverse functions.

Step 1: Algebraic Proof

To show that \(f\) and \(g\) are inverses of each other algebraically, we need to show that \(f(g(x)) = x\) and \(g(f(x)) = x\). For this step, replace \(x\) in function \(f\) by \(g(x)\) and also \(x\) in function \(g\) by \(f(x)\) and simplify the expressions in both cases. If we end up with \(x\) in both cases, then \(f\) and \(g\) are inverses of each other. Starting with \(f(g(x))\), we have \(f(g(x)) = f(8 + x^2) = -\sqrt{8 + x^2 - 8} = -\sqrt{x^2} = -x\), since \(x \leq 0\). Now, for the other direction, we have \(g(f(x)) = g(-\sqrt{x-8}) = 8 + (-\sqrt{x-8})^2 = 8 + x-8 = x\).

Step 2: Graphical Proof

To show that \(f\) and \(g\) are inverses graphically, we plot both functions on the same coordinate plane. The graph of \(g\) should be a reflection of the graph of \(f\) over the line \(y = x\). Because \(f\) and \(g\) are both defined for \(x \leq 0\), their graphs would only appear in the 2nd and 3rd quadrants. The graph of \(f\) is part of the graph of the function \(y = -\sqrt{x}\), shifted 8 units to the right, while the graph of \(g\) mirrors that of \(f\) over the line \(y = x\). This is interpreted as the function \(f\) and \(g\) being inverses.

Step 3: Numerical Proof

To give a numerical proof that \(f(g(x))=x\) and \(g(f(x))=x\), we can choose any numbers that are less than or equal to 0. For instance, when \(x = -2\), \(f(g(-2)) = -\sqrt{8 + (-2)^2 - 8} = -2\) and \(g(f(-2)) = 8 + (-\sqrt{-2-8})^2 = -2\). And when \(x = -3\), \(f(g(-3)) = -\sqrt{8 + (-3)^2 - 8} = -3\) and \(g(f(-3)) = 8 + (-\sqrt{-3-8})^2 = -3\) . Hence, \(f(g(x)) = x\) and \(g(f(x)) = x\) for these chosen values of \(x\).

Key Concepts

Algebraic Proof of Inverse Functions

Understanding how to prove two functions are inverses of each other algebraically can be a rewarding challenge. The core idea lies in showing that one function undoes the action of another. For our given functions, we must confirm that applying them sequentially returns us to our original input value. This reciprocal relationship is symbolized by the equations:

• \(f(g(x)) = x\)
• \(g(f(x)) = x\)

To illustrate this with our specific functions, we begin with \(f(g(x))\). After substitution, simplification reveals that \(f(g(x)) = -x\), given our domain of \(x \leq 0\). Proceeding similarly for \(g(f(x))\), we once again arrive at the conclusion that \(g(f(x)) = x\). Hence, algebraically, we've shown that our functions \(f\) and \(g\) are indeed inverse operations of each other.

Graphical Proof of Inverse Functions

The graphical portrayal of inverse functions offers a visual testament to their symbiotic relationship. A function and its inverse will be reflections of each other across the line \(y = x\). From a geometric perspective, this means that any point \((a, b)\) on the graph of one function, will correspond to the point \((b, a)\) on the graph of its inverse.
When we depict our functions \(f\) and \(g\) on a coordinate plane, paying special attention to their domain \(x \leq 0\), we anticipate observing this reflective property. The graph of \(f\) — a modified square root function — and the graph of \(g\), a squared function plus eight, should mirror each other along the line \(y = x\) if they are, in fact, inverses. By inspecting their graphs for this symmetry, we visually confirm what we've established algebraically.

Numerical Proof of Inverse Functions

A numerical proof adds empirical weight to our theoretical assertions, demonstrating that the functions behave as inverses under concrete conditions. In this method, potential values of \(x\), specifically those adhering to the functions' shared domain \(x \leq 0\), are substituted into the composite functions \(f(g(x))\) and \(g(f(x))\).
For example, choosing \(x = -2\) and \(x = -3\) as test cases, we observed that applying the functions in sequence — first \(g\) then \(f\), and vice versa — yielded the original values of \(x\). Through these selected values, we validate on a smaller scale what has been proven algebraically and graphically: the functions \(f\) and \(g\) are structurally designed to be inverses of one another.