Problem 35 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of Inverse Function

To find the inverse of a function, we need to find a function that 'undoes' the action of the original function. Essentially, if you start with a number, apply the function, and then apply its inverse, you should get back to the starting number.

Step 2: Write the Function as an Equation

Begin with the function given: \[ y = x^3 + 3 \] Consider it as an equation, where \( y \) is the output corresponding to the input \( x \).

Step 3: Swap Variables and Solve for x

To find the inverse, switch \( x \) and \( y \) in the equation: \[ x = y^3 + 3 \] Now, solve for \( y \). First, isolate \( y^3 \) by subtracting 3 from both sides: \[ x - 3 = y^3 \] Then, take the cube root of both sides to solve for \( y \): \[ y = \sqrt[3]{x - 3} \]

Step 4: Express the Inverse Function

Now, express the inverse function in terms of \( x \): \[ f^{-1}(x) = \sqrt[3]{x - 3} \] This inverse function will transform an output back into the input that generated it in the original function.

Step 5: Graph the Original and Inverse Functions

Graph the original function \( f(x) = x^3 + 3 \), which is a cubic function shifted 3 units up. The inverse \( f^{-1}(x) = \sqrt[3]{x - 3} \) is a cube root function shifted 3 units to the right. Both graphs should be symmetrical about the line \( y = x \).

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial function where the highest degree of the variable is three. The standard form of a cubic function is expressed as \( f(x) = ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \). The graph of a cubic function typically has a distinctive "S" shape; it may have one hump like a snake or two turning points.

Key characteristics of cubic functions include:

• They can have up to three real zeros.
• Cubic functions are continuous and smooth.
• They have one or two turning points and can change direction.

The specific function given in this exercise, \( f(x) = x^3 + 3 \), represents a basic cubic function shifted vertically upwards by three units. This shift does not affect the overall shape of the graph but moves it along the y-axis. Thus, this function retains the signature "S" shape of cubic graphs.

Function Transformation

In mathematics, function transformation refers to altering the position or shape of the graph of a function. These transformations include shifts, stretches, compressions, and reflections. A transformation shifts the graph in various directions on the coordinate plane.

For the cubic function \( f(x) = x^3 + 3 \), a vertical shift occurs. This shift is witnessed by adding 3 to the base function \( x^3 \). Such a transformation moves the whole graph upward by 3 units. Hence, every point on the graph of \( x^3 \) also moves 3 units higher.

• Vertical Shift: Adds or subtracts a constant to each y-value, impacting only the output without changing the shape.
• Horizontal Shift: Adjusts the input within the function, though not applicable directly here, it's valuable for understanding how \( f(x) = a(x-h)^3 + k \) fundamentally shifts.

These changes are vital for graphing and understanding inverse functions, each adjustment translating visually to another result.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane and connecting them to show the whole function's behavior. To graph cubic functions like \( f(x) = x^3 + 3 \), start by making a table of values, computing outputs for various inputs, and then plotting these on a grid.

For our exercise, make sure to:

• Identify critical points, such as where the graph crosses the axes.
• Note that the graph is translated upwards by 3 units from the basic \( x^3 \) graph.
• The turning point typically happens only once for this base cubic without the quadratic term.

After plotting \( f(x) \), we also graph its inverse, \( f^{-1}(x) = \sqrt[3]{x - 3} \). This function is a horizontal shift of the parent cube root function by 3 units. Check symmetry with respect to the line \( y = x \), a key feature of function and inverse graphs. This line acts as a mirror, ensuring both graphs reflect symmetrically.