Problem 49 - Elementary and Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=(x+10)^{3}\)

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[3]{x} - 10 \).

Step 1: Understand the function

The function given is \( f(x) = (x + 10)^3 \). It is a cubic function, which is known to be one-to-one, meaning it has an inverse.

Step 2: Swap \( x \) and \( f(x) \)

To find the inverse, start by replacing \( f(x) \) with \( y \), so \( y = (x + 10)^3 \). Then, swap \( x \) and \( y \) in the equation: \( x = (y + 10)^3 \).

Step 3: Solve for \( y \)

To solve for \( y \), take the cube root of both sides to get \( \sqrt[3]{x} = y + 10 \). Then, isolate \( y \) by subtracting 10 from both sides: \( y = \sqrt[3]{x} - 10 \).

Step 4: Write the inverse function

The inverse function is written as \( f^{-1}(x) \). Thus, the inverse of \( f(x) = (x + 10)^3 \) is \( f^{-1}(x) = \sqrt[3]{x} - 10 \).

Key Concepts

Cubic Functions

Understanding cubic functions can be quite exciting! These functions are polynomials of degree three, meaning the highest exponent of the variable is three. A typical form of a cubic function is given by the equation \( f(x) = ax^3 + bx^2 + cx + d \), where \(a\), \(b\), \(c\), and \(d\) are constants, and importantly, \(a\) should not equal zero.

Cubic functions have distinct characteristics that are essential to grasp:

• They can have one, two, or three real roots depending on their discriminant. However, they will always have at least one real root.
• Cubic functions are continuous and smooth, with no sharp turns.
• The shape of their graph can vary – they may have a twisting S-shape or a U-shape.
• They can exhibit inflection points where the curve changes direction concavity-wise.

The function provided in the exercise, \( f(x) = (x + 10)^3 \), uniquely lacks the more complex behavior since it's a simple case where only one term of the cubic form is present. Thus, it's straightforward to find an inverse, as it behaves predictably in its one-to-one nature.

One-to-One Functions

Now let's move on to one-to-one functions, which are crucial in determining whether a function has an inverse. Simply put, a one-to-one function means that for every value of \( y \), there is exactly one value of \( x \). In other words, no two different \( x \)-values produce the same \( y \)-value, ensuring a unique mapping. This property is also referred to as being bijective.

One way to test if a function is one-to-one is to use the horizontal line test on its graph. Draw horizontal lines across the graph; if any line touches the graph more than once, the function is not one-to-one.

The function given, \( f(x) = (x + 10)^3 \), is one-to-one. Here's why:

• Every input \( x \) is transformed into a new output by shifting \( x \) by 10 and then cubing the result. This transformation is unique for each input.
• Therefore, it qualifies as a one-to-one function, allowing us to find an inverse function that will map every unique \( y \)-value back to its original \( x \).

Recognizing this one-to-one relationship is foundational in our ability to determine inverses effectively.

Function Notation

Function notation is a concise way to represent functions and their inverses. It uses symbols like \( f(x) \) to describe the rule applied to any input \( x \) to produce the output. The notation simplifies communication about what a function does, helping us easily refer to specific operations without lengthy descriptions.

In our example, \( f(x) = (x + 10)^3 \), the notation indicates that for every \( x \), you add 10 and then cube the result. This notation captures the entire operation in just a few symbols.

When finding an inverse function, the notation shifts to \( f^{-1}(x) \), signifying that this new function undoes the operation of \( f(x) \). The inverse \( f^{-1}(x) = \sqrt[3]{x} - 10 \) essentially reverses the original process: it takes the cube root of \( x \) and subtracts 10.

Using function notation efficiently allows mathematicians and students to clearly convey complex operations and relationships in just one line. It is a powerful tool for communicating the journey from input to output and vice versa.