Question

In Exercises 23–28, find the inverse of the function. Then graph the function and its inverse. $$ f(x)=(x+4)^3 $$

Short answer

The inverse of the function \(f(x)=(x+4)^3\) is \(f^{-1}(x) = \sqrt[3]{x} - 4\). Both the function and its inverse can be graphed by selecting appropriate \(x\) values and calculating the corresponding function values.

Step-by-step solution

Finding the inverse function

Write the function in the form of \(y=f(x)\), i.e., \(y=(x+4)^3\). To find the inverse, interchange \(x\) and \(y\), giving \(x=(y+4)^3\). Now, solve for \(y\), the new equation is \(y=\sqrt[3]{x} - 4\). This is the inverse of the function.

Graphing the original function

Now, the function \(f(x)=(x+4)^3\) should be graphed. Choose a set of values for \(x\) and compute the corresponding \(f(x)\) values. The points formed from these pairs will represent the graph of \(f(x)\).

Graphing the inverse function

Similarly, for the inverse function \(f^{-1}(x) = \sqrt[3]{x} - 4\), a set of values for \(x\) will be chosen, and the corresponding \(f^{-1}(x)\) values will be calculated. The obtained points will represent the graph of \(f^{-1}(x)\).

Key concepts

Cubic Functions

Cubic functions are expressions where the highest power of the variable is three. A general form of a cubic function is represented as \(f(x) = ax^3 + bx^2 + cx + d\). Here, \(a\), \(b\), \(c\), and \(d\) are constants, with \(a eq 0\) to ensure the cubic nature of the function. These functions can take various shapes, often featuring one or two turning points, and can intersect the x-axis up to three times. An interesting property of cubic functions is their ability to model real-world phenomena that require the depiction of complex behaviors, such as certain growth patterns and physical movements. For the function \(f(x) = (x+4)^3\), it is a shifted version of the parent function \(y = x^3\). The term \(+4\) indicates a horizontal shift to the left by four units. Recognizing these transformations allows us to understand changes to a function's graph without complex calculations.

Function Graphing

Graphing functions is a powerful way to visualize what happens with different mathematical expressions. To graph \(f(x) = (x+4)^3\), you start by selecting values for \(x\), for example, \(-6, -5, -4, -3, -2\). Then, substitute these values into the function to find corresponding \(f(x)\) values. * For example, for \(x = -6\), substitute into the function: \(f(-6) = (-6+4)^3 = (-2)^3 = -8\). * Continue this process for all chosen \(x\) values. Graphing these points on a coordinate plane shows how the function behaves. The nature of the cubic function should manifest with a smooth curve, showing where the function increases and decreases. Similarly, to graph the inverse \(f^{-1}(x) = \sqrt[3]{x} - 4\), follow the same process but with this new function. The curve will show how inputs and outputs of the inverse function align differently than the original. This graphing exercise helps us comprehend the relationship between a function and its inverse through visual representation.

Inverse Relationships

Inverse functions essentially reverse the operation performed by a function. Finding an inverse requires exchanging roles of dependent and independent variables (typically \(x\) and \(y\)). For example, if we start with \(f(x)\), solving \(y = f(x)\) for \(x\) gives us the inverse, \(f^{-1}(x)\). These functions have a unique relationship: the composition of the function with its inverse returns the original input value, i.e., \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).In our case, the original function \(f(x) = (x+4)^3\) has the inverse \(f^{-1}(x) = \sqrt[3]{x} - 4\). This transformation means that each \(x\) value input into \(f(x)\) maps to a \(y\) value, which "flips back" to the original \(x\) via \(f^{-1}(x)\).Graphically, you can observe symmetry about the line \(y = x\) when plotting the original function and its inverse. This visually represents the concept of an inverse, depicting how the output of one function becomes the input of the other, creating a mirrored effect around the y=x line.