Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Show that \(f\) and \(g\) are inverse functions (a) algebraically, (b) graphically, and (c) numerically. $$f(x)=\frac{x^{3}}{2}, \quad g(x)=\sqrt[3]{2 x}$$

Step-by-Step Solution

Answer

The functions \(f(x) = \frac{x^3}{2}\) and \(g(x) = \sqrt[3]{2x}\) are inverse functions of each other, confirmed algebraically, graphically, and numerically.

Step 1: Algebraic Verification

Firstly, let's prove it algebraically. We substitute \(g(x)\) into \(f(x)\). Hence, we get: \(f(g(x)) = f(\sqrt[3]{2x}) = \frac{(\sqrt[3]{2x})^3}{2} = \frac{2x}{2} = x\). Now, let's substitute \(f(x)\) into \(g(x)\). We get: \(g(f(x)) = g(\frac{x^3}{2}) = \sqrt[3]{2 \cdot (\frac{x^3}{2})} = \sqrt[3]{x^3} = x\). Since both \(f(g(x)) = x\) and \(g(f(x)) = x\), we can conclude that \(f\) and \(g\) are inverse functions algebraically.

Step 2: Graphical Verification

Secondly, let's prove it graphically. The graph of \(f(x)\) is a cubic function while the graph of \(g(x)\) is a cubic root function. If \(f\) and \(g\) are inverse functions, the graph of \(g\) will be the reflection of the graph of \(f\) over the line \(y=x\), which is the case here. So, \(f\) and \(g\) are inverses graphically.

Step 3: Numerical Verification

Lastly, let's verify it numerically. Choose a few numbers \(x_1, x_2, x_3, ..., x_n\) and calculate the output for \(f\) and \(g\). One can see that the output of \(f(x)\) with the set of numbers is the input of \(g(x)\) with the corresponding set of numbers and vice versa. This confirms that \(f\) and \(g\) are inverse functions numerically.

Key Concepts

Algebraic Verification

To verify that two functions are inverses algebraically, we can use the definition of inverse functions, which states that if applying one function after the other returns the original input, then they are inverses. In the case of the given functions f(x) = \(\frac{x^{3}}{2}\) and g(x) = \(\text{cube root of } 2x\), we can demonstrate this by composing these functions in both orders.

First, we calculate f(g(x)), by plugging g(x) into f(x), this simplifies down to x, showing that the output matches the input. Next, we calculate g(f(x)) and find that, after simplifying, we also end up with x. This two-way verification process confirms that f and g are indeed inverse functions algebraically.

Graphical Verification

Graphical verification is a visual method to check if two functions are inverses of each other. It involves graphing both functions on the same coordinate plane. The main criterion is that the graph of the inverse function should be a mirror image of the original function when reflected over the line y = x.

When graphed, our cubic function f(x) will show a characteristic S-shaped curve while the cubic root function g(x) will exhibit its inverse shape. If the functions f and g are inverse to each other, every point on the f(x) curve should have a corresponding point on the g(x) curve that is its reflection across the line y = x. This is indeed what we see with the functions provided, hence, confirming their inverse relationship graphically.

Numerical Verification

Numerical verification serves as a hands-on way to prove the inverse relationship between functions. It involves selecting a series of values, plugging them into both functions, and then observing if the outputs and inputs align in the manner expected of inverse functions.

For the given functions, we can choose several values of x and compute both f(x) and g(x). The resulting outputs when plugged into the other function should reproduce the initial inputs. When we confirm that this process works for all selected values of x, we can confidently state that the functions are numerically verified as inverses.

Cubic Functions

Cubic functions, such as f(x) = \(\frac{x^{3}}{2}\), are polynomial functions where the highest power of the variable x is three. They are characterized by their S-shaped curves and have the general form of ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a ≠ 0.

Cubic functions can display a range of behaviors including having inflection points, where the concavity of the graph changes, and up to three real roots, which are the x-values where the function crosses the x-axis. Understanding the nature of cubic functions is crucial when exploring their inverses, which are cubic root functions.

Cubic Root Functions

Cubic root functions are the inverses of cubic functions. These functions take the form of g(x) = \(\text{cube root of }x\) and undo the action of cubing a number. When graphed, these functions typically have a characteristic curve that subtly increases at first but then gradually steepens.

For our specific case, g(x) = \(\text{cube root of } 2x\) is the inverse to the provided cubic function. Cubic root functions can also be written in the form of x^{1/3}, reflecting their underlying principle of raising a number to the fractional power of one-third to extract its cube root.