Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Show that \(f\) and \(g\) are inverse functions (a) algebraically, (b) graphically, and (c) numerically. $$f(x)=\sqrt[4]{3 x-10} ; \quad g(x)=\frac{x^{4}+10}{3}, \quad x \geq 0$$

Step-by-Step Solution

Answer

Algebraically, it is shown that \(f(g(x)) = x\) and \(g(f(x)) = x\) proving that f(x) and g(x) are inverse of each other. Graphically, g is the reflection of f across the line \(y=x\). Numerically, the tables of values for both f and g reverse in order, implying the functions are inverses. Thus through all three tests, it can be concluded that \(f(x)\) and \(g(x)\) are inversely related.

Step 1: Algebraical Proof

Set up the functions such that f(g(x)) and g(f(x)) are to be found. For f(g(x)), plug the function g(x) into f(x): \(f(g(x)) = \sqrt[4]{3(\frac{x^{4}+10}{3})-10} = x\). For \(g(f(x))\), plug the function f(x) into g(x): \(g(f(x))= \frac{( \sqrt[4]{3x-10})^{4}+10}{3} = x\). Hence, it is shown that \(f(g(x)) = x\) and \(g(f(x)) = x\), verifying algebraically that f and g are inverse functions.

Step 2: Graphical Proof

To demonstrate the functions are inverses graphically, plot the functions f and g and line \(y=x\) on the same Cartesian plane. Based on the equations, define a range of x-values and compute the corresponding y-values. Notice if the graph of g is a reflection of the graph of f across the line \(y=x\), then f and g are inverse.

Step 3: Numerical Proof

Create table of values for f and g and compare. Choose a set of x-values and find the corresponding y-values for both f and g. These ordered pairs should reverse when comparing f's set of ordered pairs to g's. If true, this would numerically confirm that the functions are inverses.

Key Concepts

Algebraic Verification

Algebraic verification of inverse functions involves demonstrating that the compositions of the functions equal the identity function. We start by finding \(f(g(x))\) and \(g(f(x))\). This means we substitute \(g(x)\) into \(f(x)\) and \(f(x)\) into \(g(x)\).

For \(f(g(x))\), if you substitute \(g(x) = \frac{x^4 + 10}{3}\) into \(f\), you get \[f(g(x)) = \sqrt[4]{3 \left(\frac{x^4 + 10}{3}\right) - 10} = \sqrt[4]{x^4} = x\].
This shows that when you input \(g(x)\) into \(f\), you simply get \(x\) back. Similarly, you need to check \(g(f(x))\).

Plug \(f(x) = \sqrt[4]{3x - 10}\) into \(g\) to get:
\[g(f(x)) = \frac{(\sqrt[4]{3x - 10})^4 + 10}{3} = \frac{3x - 10 + 10}{3} = x\].
This confirms that inputting \(f(x)\) into \(g\) also returns \(x\). Hence, \(f\) and \(g\) are algebraically verified as inverse functions.

Graphical Representation

Graphing inverse functions can visually confirm if two functions are indeed inverses. If one graph is a mirror image of the other across the line \(y = x\), then the two functions are inverses. To graph, start by plotting \(y = \sqrt[4]{3x - 10}\) and \(y = \frac{x^4 + 10}{3}\) on a coordinate plane.

Include the line \(y = x\), which acts as the mirror line. Calculate a range of \(x\)-values and find their corresponding \(y\)-values for both functions.
When you plot these points, observe whether they perfectly reflect across the line \(y = x\).

This means that any point on \(f\)'s graph for a particular \(x\)-value should correspond to a reflected point on \(g\)'s graph at the same \(y\)-value, and vice versa, confirming the inverse relationship. This graphical method helps to understand the symmetry involved with inverse functions, providing a clear visual confirmation.

Numerical Verification

Numerically verifying inverse functions involves checking if the ordered pairs essentially reverse each other when switching between the functions. First, create a set of \(x\)-values. For each \(x\), determine \(f(x)\) and \(g(x)\).

For instance, choose values like \(x = 1, 2, 3\) and calculate \(f(x)\) and \(g(f(x))\). You should find that \(g(f(x))\) returns the original \(x\) values, confirming that \(f\) maps to \(g\) and vice versa.

A table might look like this:

• For \(x = 1\), \(f(1) = \sqrt[4]{3 \times 1 - 10} = \text{not defined}\), choose valid \(x\) values like \(x = 14\): \(f(14) = 2\), \(g(2) = 14\).
• Repeat with other values to see the pattern.

By verifying that input values and their transformations mirror each other through the two functions, you affirm their inverse nature numerically. This numerical approach solidifies the understanding with actual values, thus strengthening confidence in identifying inverse functions.