Problem 57 - A Graphical Approach to College Algebra

Question

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=x^{3}+1$$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \sqrt[3]{x - 1} \); domain and range for both are all real numbers.

Step 1: Checking One-to-One Property

A function is one-to-one if each output is determined by exactly one input. A monotonic function (always increasing or decreasing) is one-to-one. Since the function \( y = x^3 + 1 \) is a cubic polynomial and has a continuously increasing slope throughout, it is a one-to-one function.

Step 2: Finding the Inverse Function

To find the inverse, start with the equation \( y = x^3 + 1 \). Swap \( x \) and \( y \) to get \( x = y^3 + 1 \). Solve for \( y \) to find the inverse: Subtract 1: \( x - 1 = y^3 \) Take the cube root: \( y = \sqrt[3]{x - 1} \) Thus, the inverse function is \( f^{-1}(x) = \sqrt[3]{x - 1} \).

Step 3: Graphing the Function and Its Inverse

Graph the original function \( y = x^3 + 1 \). This function is a cubic curve shifted up by 1 unit. It passes through points like (-1,0), (0,1), (1,2), and (2,9). Graph the inverse function \( y = \sqrt[3]{x - 1} \). This graph is the reflection of the original function across the line \( y = x \). It passes points like (0,-1), (1,0), (8, 1), and (9, 2). Both graphs intersect this line at points where the original and inverse values are equal.

Step 4: Determining Domain and Range

For the original function \( f(x) = x^3 + 1 \): - The domain is all real numbers \((-\infty, \infty)\) because a cubic function has no restrictions.- The range is also all real numbers \((-\infty, \infty)\) as it can take any real output.For the inverse function \( f^{-1}(x) = \sqrt[3]{x - 1} \):- The domain is all real numbers \((-\infty, \infty)\) as cube root can take any real input.- The range is also all real numbers \((-\infty, \infty)\) since cube root functions map to all real numbers.

Key Concepts

One-to-One Functions

Understanding one-to-one functions is key in determining if an inverse function can exist. A function is one-to-one if each output value corresponds to one unique input value. This means that no two different inputs produce the same output.
A simple way to test if a function is one-to-one is to use the horizontal line test. If any horizontal line intersects the graph at more than one point, the function is not one-to-one. However, if every horizontal line intersects the graph at most once, the function passes the test and is considered one-to-one.

• For example, the function given in the exercise, \( y = x^3 + 1 \), is a one-to-one function.
• This is because its derivative, \( 3x^2 \), is always positive for all real numbers, indicating a continuously increasing function.

This continuous increase ensures one unique output for every input, proving it is one-to-one.

Cubic Functions

Cubic functions are polynomial functions of degree three and are generally expressed in the form \( y = ax^3 + bx^2 + cx + d \). In the exercise, we have a simplified version, \( y = x^3 + 1 \), where \( a = 1 \), \( b = 0 \), \( c = 0 \), and \( d = 1 \).
These functions exhibit notable characteristics:

• They have at most 3 real roots, which are the points where the function crosses the x-axis.
• Cubic functions have one point of inflection, where the concavity of the graph changes.
• For \( y = x^3 + 1 \), the graph shifts up by one unit, making it move through the point (0,1) instead of the origin.

This particular function increases continually without bounds, indicating both domain and range span all real numbers. Its continuous nature aids in its being one-to-one, allowing an inverse to exist.

Domain and Range

Understanding the domain and range of a function gives insight into its possible input and output values. Let's delve into these concepts for the given example:

• The domain refers to the set of all possible input values (x-values) and is typically influenced by any restrictions such as divisions by zero or square roots of negative numbers.
• The range denotes the set of possible output values (y-values) that the function can produce.

For the cubic function \( f(x) = x^3 + 1 \), both the domain and range are all real numbers, \((-finity, finity)\), as it can accept any real number input and produce any real number output.
Interestingly, the inverse function \( f^{-1}(x) = \sqrt[3]{x - 1} \) also shares the same domain and range, as it is effectively only a transformed reflection of the original cubic function.

Graphing Functions

Graphing functions can provide a visual understanding of the behavior of the function. Let's explore the original function \( y = x^3 + 1 \) and its inverse \( y = \sqrt[3]{x - 1} \):
For the graph of \( y = x^3 + 1 \), imagine the familiar shape of an x-cubed graph. It will be shifted upward by 1. This transformation affects the pattern observed in its curve without changing its overall shape.

• It passes through points like (-1,0), (0,1), (1,2), and (2,9).

The inverse function \( y = \sqrt[3]{x - 1} \) is a reflection of the original function across the line \( y = x \). Graphing the inverse helps verify it:

• The curve passes through points such as (0,-1), (1,0), (8,1), and (9,2).
• Both original and inverse will intersect the line \( y = x \) at points where the same value is given in both functions.

The symmetry of the reflections confirms that these curves are true inverses of each other, visually reinforcing the analytical findings.