Problem 26 - Precalculus with Limits

Question

In Exercises 23-34, show that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. \(f(x) = 3 - 4x\), \(g(x) = \frac{3-x}{4}\)

Step-by-Step Solution

Answer

Yes, the functions \(f(x) = 3 - 4x\) and \(g(x) = \frac{3-x}{4}\) are inverses of each other both algebraically and graphically.

Step 1: Algebraically check for inverses

Plug \(g(x)\) into \(f(x)\) and simplify to see if it equals \(x\). So, calculate \(f(g(x))\). Note that \(f(g(x)) = f(\frac{3-x}{4}) = 3 - 4*(\frac{3-x}{4})\). Simplifying the right side, we find \(f(g(x)) = x\). Then plug \(f(x)\) into \(g(x)\) and simplify to see if it equals \(x\). So, calculate \(g(f(x))\). Note that \(g(f(x)) = g(3 - 4x) = \frac{3 - (3 - 4x)}{4}\). Simplifying the right side, we find \(g(f(x)) = x\). Since both :\ (f(g(x)) = x\) and \(g(f(x)) = x\), we can conclude that \(f\) and \(g\) are algebraic inverses of each other.

Step 2: Graphically check for inverses

To graphically determine if \(f\) and \(g\) are inverses, one needs to plot both functions and the line \(y = x\) on the same graph. The functions \(f\) and \(g\) are inverses of each other if and only if the graph of \(g\) is a reflection of the graph of \(f\) in the line \(y = x\). In this case, plotting these equations on a graph will show that indeed, the graph of \(g\) is a reflection of the graph of \(f\) in the line \(y = x\). Therefore, \(f\) and \(g\) are also graphically inverses.

Step 3: Conclusion

Since \(f\) and \(g\) have both met the conditions for being inverses of each other algebraically and graphically, we can conclude they are indeed inverse functions.

Key Concepts

Algebraic Verification of Inverses

Verifying inverse functions algebraically involves demonstrating that each function cancels out the other, specifically when composed together. This means if you take function \(g\) and substitute it into function \(f\), the result should be the original input \(x\), and vice-versa. Here is how you can check:

• Start by calculating \(f(g(x))\). You substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = f\left(\frac{3-x}{4}\right) = 3 - 4\left(\frac{3-x}{4}\right)\).
• By simplifying this expression, \(3 - (3-x)\), results in \(x\) indicating \(f(g(x)) = x\).
• Now, find \(g(f(x))\). Substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g(3 - 4x) = \frac{3 - (3 - 4x)}{4}\).
• Simplifying gives \(\frac{3 - 3 + 4x}{4} = \frac{4x}{4} = x\), which shows \(g(f(x)) = x\).

Both compositions result in \(x\), thus proving \(f\) and \(g\) are inverses. This method assures us that the two functions reverse each other’s actions completely.

Graphical Verification of Inverses

Visualizing inverse functions using graphs involves plotting both functions and examining their reflective relationship. It's an easy and effective way to verify if two functions are inverses.

• Plot the graph of \(f(x) = 3 - 4x\).
• Next, plot the graph of \(g(x) = \frac{3-x}{4}\).
• Also, draw the line \(y = x\) on the same graph. This line, \(y = x\), acts as the axis of symmetry in the context of inverses.

The graphs of \(f\) and \(g\) should lie on opposite sides of the line \(y = x\) but be mirror images relative to this line. If you fold the graph along \(y = x\), the graph of \(g\) should coincide with that of \(f\), showing they reflect across it. Thus, confirming function \(f\) and \(g\) are indeed graphically inverse functions.

Reflection Across \(y = x\)

When two functions are inverses, graphically they reflect across the diagonal line \(y=x\), making this reflection an important characteristic to understand.

• The line \(y = x\) plays a central role by acting as a mirror between the graphs of two inverse functions.
• If a point \((a, b)\) lies on the graph of \(f\), then \((b, a)\) will lie on the graph of \(g\), the inverse function.
• As a result, each point on function \(f\) switches its \(x\) and \(y\) values to become a point on function \(g\), illustrating their reflection over the line \(y = x\).

Understanding this reflection property helps in recognizing inverse relationships visually. The reflection not only provides a solution but also builds intuition for identifying similar patterns within other graphical functions.