Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Show that \(f\) and \(g\) are inverse functions (a) algebraically, (b) graphically, and (c) numerically. $$f(x)=x^{3}+5, \quad g(x)=\sqrt[3]{x-5}$$

Step-by-Step Solution

Answer

The functions \(f(x)=x^3+5\) and \(g(x)=\sqrt[3]{x-5}\) are indeed inverses of each other. This conclusion is based on our algebraic manipulation which yielded \(f(g(x)) = x\) and \(g(f(x)) = x\), which is the basic property of inverse functions. Their graphical representations would also reflect each other in the line \(y=x\), and a numerical check would show corresponding values across the functions.

Step 1: Proving Algebraically

To prove two functions are inverse algebraically, we need to show that \(f(g(x)) = x\) and \(g(f(x)) = x\). \Substitute \(g(x)\) into \(f\): \\[f(g(x))= (\sqrt[3]{x-5})^3+5 = x. \]And then substitute \(f(x)\) into \(g\) : \\[g(f(x))= \sqrt[3]{(x^3+5-5)} = x.\]

Step 2: Proving Graphically

We can visually confirm that the two functions are inverses by plotting them on the same graph. The graph of \(f(x)\) should be a reflection of the graph of \(g(x)\) about the line \(y=x\). If they are, it means they are inverse functions.

Step 3: Proving Numerically

To prove inverse numerically, you create a table of values for \(f\) and \(g\). If \(f\) and \(g\) are inverses, every input of \(f\) corresponds to an output of \(g\), and vice versa. This means that the x-values of function \(f\) should be the y-values of the function \(g\) and the x-values of function \(g\) should be the y-values of the function \(f\). This can be checked by establishing a series of values for x, finding the corresponding y-values for each function, and checking to see if the required condition is fulfilled.

Key Concepts

Algebraic Proof

When it comes to proving that two functions are inverses of each other algebraically, the key is to ensure that they can reverse each other’s operations. For the functions \( f(x) = x^3 + 5 \) and \( g(x) = \sqrt[3]{x - 5} \), this means verifying two conditions: whether \( f(g(x)) = x \) and \( g(f(x)) = x \).

• To find \( f(g(x)) \), substitute \( g(x) \) into \( f \): \[ f(g(x)) = \left( \sqrt[3]{x - 5} \right)^3 + 5. \] Simplifying inside the parenthesis eliminates the cube root with the cube, resulting in \( x - 5 \), and adding 5 resets it to \( x \).
• Similarly, substitute \( f(x) \) into \( g \) to find \( g(f(x)) \): \[ g(f(x)) = \sqrt[3]{(x^3 + 5) - 5} = \sqrt[3]{x^3}. \] The cube and the cube root again cancel each other, returning \( x \).

Both calculations affirm that \( f \) and \( g \) are indeed inverse functions, as substituting one into the other yields \( x \), demonstrating an elegant symmetry in their operations.

Graphical Representation

A visual way to verify if two functions are inverses involves graphing both functions on the same plane and checking their symmetry about the line \( y = x \). This line acts almost like a mirror.

• Graph \( f(x) = x^3 + 5 \), which shifts the usual cubic graph upwards by 5 units.
• Graph \( g(x) = \sqrt[3]{x - 5} \), which shifts the typical cubic root graph to the right by 5 units.

When plotted, the graphs for \( f \) and \( g \) should mirror each other across the line \( y = x \). This symmetry is a tell-tale sign of inverse functions. Any point \( (a, b) \) on \( f(x) \) would correspond to a point \( (b, a) \) on \( g(x) \), confirming their inverse relationship visually.

Numerical Verification

Numerically verifying inverse functions through tables of values can make their inverse relationship more tangible. This process involves computing a series of \( x \) values and comparing outputs.

• Choose sample input values such as \( x = 1, 2, 3, \ldots \).
• Compute \( f(x) \) for these values to get outputs (\( y \)-values).
• For the same \( y \)-values, calculate \( g(y) \) and check if it returns the original \( x \) values.

For instance, if \( f(1) = 6 \), then \( g(6) = 1 \) should hold true. By checking these conversions, you can numerically authenticate that \( f \) and \( g \) reciprocate each other’s operations, further validating their inverseness.