Problem 66 - Intermediate Algebra

Question

Find the inverse of each function. Then graph the function and its inverse on one coordinate system. Show the line of symmetry on the graph. $$ f(x)=x^{3}+1 $$

Step-by-Step Solution

Answer

The inverse of \( f(x) = x^3 + 1 \) is \( f^{-1}(x) = \sqrt[3]{x - 1} \). The line of symmetry is \( y = x \).

Step 1: Understanding the Function

The function provided is \( f(x) = x^3 + 1 \). It is a cubic function, which is continuous and one-to-one, meaning it should have an inverse.

Step 2: Replace f(x) with y

To find the inverse, first replace \( f(x) \) with \( y \). So, the equation becomes \( y = x^3 + 1 \).

Step 3: Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. Thus, \( x = y^3 + 1 \).

Step 4: Solve for y

Solve the equation \( x = y^3 + 1 \) for \( y \). First, subtract 1 from both sides to get \( x - 1 = y^3 \). Then, take the cube root of both sides: \( y = \sqrt[3]{x - 1} \). Thus, the inverse function is \( f^{-1}(x) = \sqrt[3]{x - 1} \).

Step 5: Graph the Original and Inverse Functions

Plot the function \( f(x) = x^3 + 1 \) and its inverse \( f^{-1}(x) = \sqrt[3]{x - 1} \) on the same coordinate system. The graph of the original function is a cubic curve shifted up by 1 unit. The inverse is a cube root curve shifted right by 1 unit.

Step 6: Draw the Line of Symmetry

Draw the line of symmetry, which is the line \( y = x \). This line reflects each point on \( f(x) \) to \( f^{-1}(x) \) and vice versa on the graph.

Key Concepts

Cubic Functions

A cubic function is a polynomial function of the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). In the exercise, the provided cubic function is \( f(x) = x^3 + 1 \). This particular function is quite simple, as it involves no \( x^2 \) or \( x \) terms, only a constant added to the cubic term.
Cubic functions are interesting because they can exhibit a variety of shapes depending on their coefficients. However, in our case, because the function is in the form \( x^3 + 1 \), it represents a smooth curve that starts from bottom left and rises to top right, making it a continuous and one-to-one function. This means each input \( x \) has a distinct output \( y \), ensuring an inverse function exists.
Some common characteristics of cubic functions like \( f(x) = x^3 + 1 \) include:

• They are always continuous and have no breaks.
• They can have one real root at minimum.
• The graph can change concavity, displaying an inflection point where it turns from curving one way to the opposite.

Understanding these attributes helps in comprehending why such functions are suitable for finding inverses, especially when the cubic is both injective (one-to-one) and surjective (onto) within the limits explored.

Graphing Inverse Functions

Graphing inverse functions requires understanding that the inverse function essentially "undoes" the original function. In the given exercise, the inverse of \( f(x) = x^3 + 1 \) is \( f^{-1}(x) = \sqrt[3]{x - 1} \). Here’s how to approach the graphing of this function and its inverse:
To graph both the original cubic function and its inverse on the same coordinate system:

• Plot the original function \( f(x) = x^3 + 1 \). Since it’s a cubic curve adjusted vertically by 1 unit, start by graphing the basic \( x^3 \) curve, then shift it upward to incorporate the constant.
• Plot the inverse function \( f^{-1}(x) = \sqrt[3]{x - 1} \). This is a cube root curve, which means it starts horizontally and then slopes upward. In this function, the horizontal shift is 1 unit to the right.

Graphing these two functions on the same coordinate system helps to visualize how the composition of the function and its inverse essentially maps each \( x \) back to the original \( f(x) \). Each point \((a, b)\) on \( f(x) \) has a corresponding point \((b, a)\) on \( f^{-1}(x) \). This is significant for understanding the relationship and symmetry that exists between them.

Line of Symmetry in Functions

The relationship between a function and its inverse can be seen clearly when both are reflected across a specific line known as the line of symmetry. For functions and their inverses, this line is usually \( y = x \). In the exercise provided, after obtaining the inverse function, drawing the line of symmetry helps to confirm the correctness of your graphical representation.
Among the key features of the line of symmetry \( y = x \) in this context are:

• It acts as a mirror line where every point on the graph of \( f(x) \) has its reflection on the graph of \( f^{-1}(x) \).
• The line \( y = x \) is a diagonal that cuts through the origin at a 45-degree angle, dividing the plane into two equal halves.

When graphing, ensure to draw this line clearly to see how each corresponding point on \( f(x) \) maps directly across to its inverse. This reflection property offers a visual proof of the mathematically correct nature of the inverse relationship.