Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 17-20: Use the theorem on inverse functions to prove that \(f\) and \(g\) are inverse functions of each other, and sketch the graphs of \(f\) and \(g\) on the same coordinate plane. $$ f(x)=x^{3}-4 ; \quad g(x)=\sqrt[3]{x+4} $$

Step-by-Step Solution

Answer

Functions \(f\) and \(g\) are inverses, verified via composition. Their graphs reflect over \(y=x\).

Step 1: Verify the Composition of Functions

To confirm that two functions are inverses, we need to check if their compositions yield the identity function. First, compute \( (f \circ g)(x) = f(g(x)) = f(\sqrt[3]{x+4}) \).Plug \(g(x)\) into \(f(x)\):\[f(\sqrt[3]{x+4}) = (\sqrt[3]{x+4})^3 - 4 = x + 4 - 4 = x.\]This shows that \((f \circ g)(x) = x\).

Step 2: Verify the Reverse Composition of Functions

Next, compute the reverse composition \( (g \circ f)(x) = g(f(x)) = g(x^3 - 4) \).Plug \(f(x)\) into \(g(x)\):\[g(x^3 - 4) = \sqrt[3]{(x^3 - 4) + 4} = \sqrt[3]{x^3} = x.\]This shows that \((g \circ f)(x) = x\).

Step 3: Conclude That Functions Are Inverses

Since both compositions, \((f \circ g)(x)\) and \((g \circ f)(x)\), return \(x\), the functions \(f\) and \(g\) are indeed inverse functions of each other.

Step 4: Sketch the Graphs of the Functions

Draw the graph of \(f(x) = x^3 - 4\), which is a cubic function shifted downward by 4 units. It will pass through the point \((0, -4)\) and will have the typical cubic shape.For \(g(x) = \sqrt[3]{x+4}\), draw a cube root graph shifted left by 4 units, passing through \((-4, 0)\) and exhibiting the symmetry and shape of a cube root function.Both graphs will reflect over the line \(y = x\), as they are inverses.

Key Concepts

Composition of Functions

Functions can be composed, meaning that you can feed the output of one function right into another. For functions to be inverses, their compositions must result in the identity function. The identity function is what you think — it just returns the original input for all arguments. To check if functions are inverses, we perform the operation

• Compute \((f \circ g)(x)\) which means you apply function \(g\) first, then \(f\).
• Compute \((g \circ f)(x)\) which means you apply function \(f\) first, then \(g\).
• Both results should be equal to \(x\).

In the given exercise, both combinations of the functions \(f(x) = x^3 - 4\) and \(g(x) = \sqrt[3]{x+4}\) return \(x\), confirming they are inverses. The process of checking composition serves as a creative way to affirm this beautiful mathematical relationship.

Cubic Function

A cubic function is any function of the form \(f(x) = ax^3 + bx^2 + cx + d\). The exercise provides us with a simple version: \(f(x) = x^3 - 4\), which is a cubic function shifted downward by 4 units along the y-axis. Key aspects of cubic functions include:

• They have a characteristic shape known as an "S" curve, with one inflection point.
• The graph of a cubic function can be symmetrical about its inflection point.
• It stretches endlessly upwards and downwards due to its power terms.

The shift of 4 units downward, evident in \(f(x)\), moves the "S" to start at \((0, -4)\) on the graph. This gives it a distinctive pivot point, adding to the visual understanding of cubic transformations.

Cube Root Function

The cube root function is essentially the inverse of the cubic function. Rather than cubing a number, it retrieves the original value that was cubed. In the exercise, the function \(g(x) = \sqrt[3]{x+4}\) shifts the cube root function horizontally to the left by 4 units. Key points to understand the cube root function include:

• They are characterized by their gentle slope compared to square root functions.
• The graph passes through the line \(y = x\), which means for every value on the graph, the input and output are symmetric.
• Like cubic functions, they extend infinitely in both vertical directions.

This transformation aligns the function to pass through the point \((-4, 0)\). Recognizing how shifts and transformations reflect on graphs helps build a stronger grasp of inverse relationships, reinforcing their visual symmetry over the line \(y = x\).