Problem 46 - College Algebra

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 \(f^{-1}\). $$ f(x)=x^{3}+1 $$

Step-by-Step Solution

Answer

a. The inverse of the function \(f(x)\), \(f^{-1}(x)\), is \((x-1)^{1/3}\). b. The graph of both functions show that they are reflections over the line \(y=x\). c. The domain and range of both \(f(x)\) and \(f^{-1}(x)\) are \((-\infty, \infty)\).

Step 1: Find the Inverse

To find the inverse function \(f^{-1}(x)\), start by replacing \(f(x)\) with \(y\). So we have \(y = x^{3} + 1\). Then swap \(x\) and \(y\) to get \(x = y^{3} + 1\). Now we need to solve this equation for \(y\) to find the inverse. Subtract 1 from both sides to have \(x-1 = y^{3}\). Obtain \(f^{-1}(x)\) as \(y = (x-1)^{1/3}\).

Step 2: Graph \(f\) and \(f^{-1}\)

Using a graphing tool, graph the function \(f(x) = x^{3} + 1\) and its inverse function \(f^{-1}(x) = (x-1)^{1/3}\). Remember that the graph of the inverse function is a reflection of the graph of the original function over the line \(y = x\).

Step 3: Determine Domains and Ranges

Now, we can find the domain and range of \(f(x)\) and \(f^{-1}(x)\). From the graph, we see that \(f(x)\) is defined for all real numbers so the domain of \(f(x)\) is \((-\infty, \infty)\). As \(f(x)\) has an output of all real numbers the range of \(f(x)\) also is \((-\infty, \infty)\). Similarly, from the graph, we see that \(f^{-1}(x)\) is defined for all real numbers so the domain of \(f^{-1}(x)\) is \((-\infty, \infty)\). As \(f^{-1}(x)\) has an output of all real numbers the range of \(f^{-1}(x)\) also is \((-\infty, \infty)\).

Key Concepts

Domain and Range

When discussing functions, the concepts of domain and range are fundamental. The domain of a function is all the possible input values (x-values) that the function can accept. For the function given in our exercise,

• The domain of \(f(x) = x^3 + 1\) includes all real numbers, denoted as \((−∞, ∞)\).

This indicates that you can plug any real number into \(f(x)\) and it will produce a real output. Similarly, the range of a function is all the possible output values (y-values) that the function can produce.

• For \(f(x) = x^3 + 1\), the range is also all real numbers \((−∞, ∞)\).

This occurs because a cubic function extends infinitely in both the positive and negative y-directions.
When we find the inverse of a function, the domain and range essentially "swap". Thus, the inverse function, \(f^{-1}(x) = (x - 1)^{1/3}\), retains the domain and range of all real numbers for both aspects. This is a key point with inverse functions: they reverse the roles of inputs and outputs, which is reflected in their domain and range outcomes.

Graphing Functions

Graphing functions visually displays the relationship between inputs and outputs, helping to better understand and solve problems. For the function \(f(x) = x^3 + 1\), we consider it as a transformation of the basic cubic function \(x^3\). The "+1" shifts the entire graph upwards by 1 unit on the Cartesian plane.
When graphing \(f(x)\), observe the following:

• Cubic functions look like an elongated "S" curve, where the left side heads down to negative infinity, the middle section passes through the origin, and the right side climbs to positive infinity.
• The graph of \(f(x)\) will cross the y-axis at \(x = 0\), resulting in \(f(0) = 1\).

When plotting the inverse, \(f^{-1}(x) = (x-1)^{1/3}\), the curve will reflect similar characteristics of an inverted "S" shape, reading and plotting the graph's behavior horizontally instead of vertically.

Reflection of Functions

Reflection is a key concept when discussing inverse functions. A function \(f(x)\) and its inverse \(f^{-1}(x)\) are reflections of each other across the line \(y = x\). This line acts as a mirror.
Imagine folding your graph paper along the line \(y = x\); the graphs of \(f(x)\) and \(f^{-1}(x)\) will lay over each other. Understanding this reflection helps:

• Verify the correctness of the inverse function by checking its symmetry with the original function.
• Ensure that points (a, b) on \(f(x)\) correspond to (b, a) on \(f^{-1}(x)\).

It's a visual confirmation of the fundamental property of inverses: outputs and inputs switch roles. If you connect points on \(f(x)\) to their corresponding points on \(f^{-1}(x)\) directly through the line \(y = x\), they should align perfectly.