Problem 47 - Precalculus Essentials

Question

a. Find an equation for \(f^{-1}(x)\) b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\) $$f(x)=(x+2)^{3}$$

Step-by-Step Solution

Answer

The inverse function of \(f(x)=(x+2)^3\) is \(f^{-1}(x)=\sqrt[3]{x}-2\). Both \(f\) and \(f^{-1}\) can be graphed on the same rectangular coordinate system, with both their domain and ranges being all real numbers, denoted as \((-∞, ∞)\).

Step 1: Find an equation for \(f^{-1}(x)\)

We start with the function \(f(x)=(x+2)^{3}\). We replace \(f(x)\) with \(y\), giving us \(y=(x+2)^{3}\). Now, we swap \(x\) and \(y\) to obtain \(x=(y+2)^{3}\). Solve this equation for \(y\). By taking the cube root of both sides, we get \(y=\sqrt[3]{x}-2\). Therefore, the inverse of the function is \(f^{-1}(x)=\sqrt[3]{x}-2\).

Step 2: Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system

Having established the equations for \(f\) and \(f^{-1}\), plot these on the same graph. Note that the graph of \(f\) is a rightwards-opening cubic and \(f^{-1}\) is a upwards-opening cube root function. Additionally, these two functions are symmetric about the line \(y=x\).

Step 3: Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\)

For the function \(f(x)=(x+2)^3\), the domain is all real numbers, so in interval notation, it is \((-∞, ∞)\). The range is also all real numbers, which is \((-∞, ∞)\) as well. For its inverse \(f^{-1}(x)=\sqrt[3]{x}-2\), the domain is also all real numbers, \((-∞, ∞)\), and the range is also all real numbers, \((-∞, ∞)\).

Key Concepts

Cubic Functions

Cubic functions represent equations of the form \( f(x) = ax^3 + bx^2 + cx + d \), where \(a, b, c,\) and \(d\) are constants with \(a eq 0\). In this category of functions, the highest degree of the polynomial is three. This makes cubic functions particularly interesting because of their distinct S-shaped curves on graphs.
An important feature of cubic functions like \(f(x) = (x+2)^3\) is their ability to model real-world phenomena with changes in direction. Understanding these functions involves recognizing their behavior, which can include turning points and inflection points, where the concavity shifts.
Identifying the key characteristics is beneficial:

• The coefficients determine the shape and direction of the graph.
• Cubic functions can intersect the x-axis up to three times.
• They always have one real root, and depending on the discriminant, may have complex roots as well.

To solve or manipulate such functions, algebra and calculus techniques, such as finding derivatives and evaluating roots, are useful tools.

Graphing Functions

Graphing techniques allow visualization of functions to better understand their behavior. For cubic functions like \(f(x) = (x+2)^3\), the graph is a smooth curve that extends from negative to positive infinity, passing through critical points.
When sketching such a function:

• Identify the vertex or turning point. For \((x+2)^3\), shifting happens at \(x=-2\).
• Calculate additional points for better accuracy by choosing values of \(x\), substituting them in the function formula.
• Recognize symmetry around points or axes, which helps anticipate the curve's appearance. Cubics show inflection symmetries.

While plotting an inverse function \(f^{-1}(x) = \sqrt[3]{x} - 2\), note its property of reflecting over the line \(y=x\). This line acts like a mirror for functions and their inverses.
Both graphs together provide insight into the correlation between original functions and their inverses, highlighting their symmetry and intersecting properties, like how they swap domain and range.

Domain and Range

Understanding domain and range is crucial in analyzing functions. The domain includes all possible input values \(x\) that a function can accept without causing errors, while the range encompasses all output values \(y\) the function can produce.
For the cubic function \(f(x)=(x+2)^3\):

• Domain: Since there are no restrictions, any real number can be plugged into \(x\), so the domain is \((-\infty, \infty)\).
• Range: The function can output any real number as well, making its range also \((-\infty, \infty)\).

For its inverse function \(f^{-1}(x)=\sqrt[3]{x}-2\):

• Domain: Much like the original function, its domain is all real numbers \((-\infty, \infty)\).
• Range: The range, consistent with the domain of \(f\), spans all real numbers \((-\infty, \infty)\).

The role of domain and range emphasizes how functions and their inverses swap these elements. This understanding aids in setting boundaries for graphing and practical applications.

Interval Notation

Interval notation is a mathematical shorthand used to define the sets of numbers, particularly useful for expressing domain and ranges. It concisely shows continuous groups of numbers, commonly seen in calculus and higher-level math.
For the function \(f(x) = (x+2)^3\) and its inverse \(f^{-1}(x) = \sqrt[3]{x} - 2\), the notation looks like:

• Domain: Expressed as \((-\infty, \infty)\), meaning every real number is included.
• Range: Also \((-\infty, \infty)\), similar to the domain, reflects the function’s behavior of covering a real number scale.

In working with interval notation:

• Brackets \([a, b]\) indicate inclusivity of endpoints \(a\) and \(b\).
• Parentheses \((a, b)\) show that endpoints aren't part of the set.

Using interval notation ensures clarity when communicating the span of values a function can handle or produce, making it an indispensable tool in mathematical expressions.