Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

On the same set of axes, sketch the graphs of the following pairs of functions. In parts (a) and (b) find an expression for \(f^{-1}(x)\). The graphs of \(f\) and \(f^{-1}(x)\) are mirror images over the line \(y=x\) since the roles of input and output are switched to obtain the inverse function. In other words, if \((1,5)\) is a point on the graph of \(f\), then \((5,1)\) is point on the graph of \(f^{-1}(x)\). (a) \(f(x)=2 x+1\) and \(f^{-1}(x)\) (b) \(f(x)=x^{2}-2, x>0\) and \(f^{-1}(x)\) (c) \(f(x)=10^{x}\) and \(f^{-1}(x)\) (d) \(f(x)=2^{-x}\) and \(f^{-1}(x)\)

Step-by-Step Solution

Answer

The inverse functions for the given functions are as follows: \(f^{-1}(x) = \frac{x - 1}{2}\), \(f^{-1}(x) = \sqrt{x + 2}\), \(f^{-1}(x) = \log_{10}(x)\), and \(f^{-1}(x) =- \log_{2}(x)\). The graphs of these functions pairs show symmetry about the line \(y = x\).

Step 1: Compute and graph \(f(x)\) and \(f^{-1}(x)\) for part (a)

The inverse of a function can be found by switching x and y. Here, for the function \(f(x) = 2x + 1\), replacing \(x\) with \(y\), we get \(x = 2y + 1\). Isolating \(y\) now, we obtain \(f^{-1}(x) = \frac{x - 1}{2}\). Graph these two functions on the same set of axes and draw the line \(y = x\). Notice how \(f\) and \(f^{-1}(x)\) are mirror images about the line \(y = x\).

Step 2: Repeat Step 1 for part (b)

The function is \(f(x) = x^2 -2\), for \(x > 0\). Swapping \(x\) and \(y\), we get \(x = y^2 - 2\). Solving for \(y\), that is the square root of the equation, we get \(f^{-1}(x) = \sqrt{x + 2}\). Graph these two functions on the same set of axes and draw the line \(y = x\). Observe how \(f\) and \(f^{-1}(x)\) are mirror images about the line \(y = x\).

Step 3: Repeat Step 1 for part (c)

The given function is \(f(x) = 10^x\). Its inverse function is obtained by swapping \(x\) and \(y\) to get \(x = 10^y\). To isolate \(y\), giving \(f^{-1}(x) = \log_{10}(x)\). Graph these two functions on the same set of axes and draw the line \(y = x\). Again, notice how the two functions are mirror images of each other about the line \(y = x\).

Step 4: Repeat Step 1 for part (d)

The function given is \(f(x) = 2^{-x}\). Swapping \(x\) and \(y\) results in \(x = 2^{-y}\). Isolating \(y\), we get \(f^{-1}(x) =- \log_{2}(x)\). Graph these two functions on the same set of axes and draw the line \(y = x\). Once again, examine how the functions are mirror images about the line \(y = x\).

Key Concepts

Function Composition

Function composition involves combining two functions into one, acting like a mathematical pipeline. Imagine you have two functions, say \(f\) and \(g\). Function composition is expressed as \((f \circ g)(x) = f(g(x))\).
It's like feeding the output of \(g(x)\) into \(f\) as its input.
This operation helps to simplify complex problems by breaking them into smaller, manageable parts.
When dealing with inverse functions, composition plays an important role.

• For instance, if \(f\) has an inverse \(f^{-1}\), then function composition helps check correctness:
• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

These equations confirm that performing an operation followed by its inverse gets you back to the beginning.
Function composition is crucial for verifying inverse functions since the original and its inverse "undo" each other.

Graphing Functions

Graphing functions helps us visualize math functions on a coordinate plane. It provides clear insights into how functions behave and relate to each other.
Consider the following steps to graph a function:

• **Identify critical points**: For linear functions like \(f(x) = 2x + 1\), plot points such as where the line crosses the y-axis.
• **Find key features**: Note features like intercepts or where a parabola vertex is if dealing with quadratic functions.
• **Draw the curve**: Connect points smoothly reflecting function nature, ensuring easy understanding of behaviors such as slopes.

For inverse functions, graphing is particularly fascinating:

• Graph \(f(x)\) and \(f^{-1}(x)\) onto the same axes.
• Graph them relative to the identity line \(y = x\).
• Notice the symmetry where both graphs mirror each other, indicating that inputs and outputs swap in the inverse.

This visualization technique is effective in checking if an inverse is correctly found, as it showcases their reflective property.

Reflection Over the Line y=x

Reflection over the line \(y=x\) is a powerful concept when dealing with inverse functions. This line, often called the identity line, helps us "see" how two functions are linked.
In geometric terms, if \((a, b)\) is on \(f\), then its reflected point \((b, a)\) is on \(f^{-1}\).
To understand this reflection better:

• Consider graphing a point like \((1,5)\).
• Reflect the point across the line \(y = x\) so it becomes \((5,1)\).

This mirroring confirms that inputs swap with outputs in inverses. When the graphs of \(f(x)\) and \(f^{-1}(x)\) are compared, their reflections over \(y = x\) highlight their relationship.
Functions and their inverses will always reflect across this specific line, providing a visual litmus test to check inverse function calculations.