Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=1-x^{3} $$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \sqrt[3]{1-x} \). It is symmetric to the original function about the line \( y = x \).

Step 1: Replace f(x) with y

Start by expressing the function as an equation with y: \[ y = 1 - x^3 \]

Step 2: Swap x and y

To find the inverse, we need to switch the roles of x and y:\[ x = 1 - y^3 \]

Step 3: Solve for y

Isolate y in the equation. First, subtract 1 from both sides:\[ x - 1 = -y^3 \]Then multiply through by -1:\[ 1 - x = y^3 \]Finally, take the cube root of both sides:\[ y = \sqrt[3]{1-x} \]This is the inverse function: \[ f^{-1}(x) = \sqrt[3]{1-x} \]

Step 4: Graph the Original Function

Graph the original function \( f(x) = 1 - x^3 \). It is a cubic curve that decreases as x increases.

Step 5: Graph the Inverse Function

Graph the inverse function \( f^{-1}(x) = \sqrt[3]{1-x} \). It is a curve that is symmetric to the original function with respect to the line \( y = x \).

Step 6: Verify by Composition

Ensure the correctness by checking \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). For \( f(f^{-1}(x)) \):\[ f(f^{-1}(x)) = 1 - (\sqrt[3]{1-x})^3 = x \]And for \( f^{-1}(f(x)) \):\[ f^{-1}(f(x)) = \sqrt[3]{1-(1-x^3)} = x \]Both compositions return the input \( x \), confirming that the functions are indeed inverses.

Key Concepts

Cubic Function

Cubic functions are polynomial functions of degree three, and they take the general form of \( f(x) = ax^3 + bx^2 + cx + d \). Here, each term represents a part of the cubic curve's shape:

• \( ax^3 \) impacts the cubic's overall direction.
• \( bx^2 \) contributes to the curve's bends and turns.
• \( cx \) adjusts the slope and tilt of the graph.
• \( d \) shifts the whole graph up or down.

The function we're looking at is \( f(x) = 1 - x^3 \). It's a specific cubic function where all coefficients but \( a \) are absent except the constant term \( d \). This simplifies the analysis, highlighting the cubic term's influence.
One characteristic feature of cubic functions is their distinct S-shaped graph, called an inflection point. Other key points are the roots, where the function crosses the x-axis, and the y-intercept, where it crosses the y-axis. In our case, because of the negative \( x^3 \) term, the curve starts high and ends low, different from a typical upward trend of a positive cubic.

Graphing Functions

Graphing functions helps us visualize mathematical expressions. It shows how inputs relate to outputs. Consider the original function, \( f(x) = 1 - x^3 \). To graph it, follow these steps:

• Identify the basic shape, using known properties of cubic functions. It involves an S-curve.
• Determine the y-intercept, found by setting \( x = 0 \). Here, the y-intercept is at \( (0,1) \).
• Find important points such as when the function hits the x-axis. Calcualte by solving \( 1 - x^3 = 0 \) for roots.

When graphing the inverse, \( f^{-1}(x) = \sqrt[3]{1-x} \), note its symmetry compared to \( f(x) \). It should reflect the original curve across the line \( y = x \).

For effective graphing, sketch the line \( y = x \) as a guide. This mirror behavior is key for identifying inverses. By plotting points and observing this symmetry, you learn how functions relate visually.

Function Composition

Function composition is the process of applying one function to the results of another. In formal terms, if \( f \) and \( g \) are functions, the composition \( f(g(x)) \) is calculated by plugging the output of \( g(x) \) into \( f \). This helps in understanding combined effects or "covered" behaviors.

In the case of inverses, the key test of whether two functions are inverses is by checking that their compositions lead back to the original input \( x \).

• Compose \( f(f^{-1}(x)) \) to ensure the result is \( x \) again. It means \( f \) "undoes" changes made by \( f^{-1} \).
• Similarly, do \( f^{-1}(f(x)) = x \) to confirm that \( f^{-1} \) "reverses" \( f \).

Checking both compositions is crucial. It confirms the functions genuinely reverse one another. This process seals the relationship between original and inverse functions, making it a vital tool in higher mathematics. Observing these interactions deepens understanding of how functions interrelate.