Problem 21 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, analytically show that the functions are inverse functions. Then use a graphing utility to show this graphically. $$ \begin{aligned} &f(x)=e^{2 x-1} \\ &g(x)=\frac{1}{2}+\ln \sqrt{x} \end{aligned} $$

Step-by-Step Solution

Answer

The functions \(f(x) = e^{2x-1}\) and \(g(x) = \frac{1}{2}+\log \sqrt{x}\) are indeed inverse functions of each other both analytically and visually on a graph.

Step 1: Analytical Verification

To analytically verify that two functions are inverses of each other, one must show that \(f(g(x)) = x\) and \(g(f(x)) = x\).Let's start with \(f(g(x))\): \[ f(g(x)) = f\left(\frac{1}{2}+\ln \sqrt{x}\right)= e^{2\left(\frac{1}{2}+\ln \sqrt{x}\right)-1}= e^{\ln x} = x\] Then \(g(f(x))\): \[ g(f(x)) = g\left(e^{2x-1}\right)= \frac{1}{2}+\ln \sqrt{e^{2x-1}} = \frac{1}{2}+ \frac{1}{2}(2x-1)=x\] Both computations yield \(x\), therefore \(f(x)\) and \(g(x)\) are inverse functions of each other.

Step 2: Graphical Verification

When two functions are inverse of each other their plots are symmetrical along the line \(y=x\). After drawing the graph of \(f(x)\), \(g(x)\), and the line \(y=x\) on the same coordinate plane, one should observe the graphs of \(f(x)\) and \(g(x)\) are reflections of each other in the line \(y=x\), thus showing that \(f\) and \(g\) are indeed inverse functions.

Key Concepts

Analytical Verification of Inverse Functions

Understanding how to verify the inverse relationship between two functions is an important mathematical skill. Analytically, this involves proving that when one function is applied to the other, the result is the original input value.
This process involves two critical steps. First, we compute the composition of the first function with the second to see if the output is the variable itself, which is often denoted as the identity function. In this case, if you take the function f(x) = e^{2x - 1} and apply its proposed inverse g(x) = \(\frac{1}{2} + \text{ln} \text{\textbackslash}sqrt{x}\), the result should be x.

• When we compute f(g(x)), we substitute g(x) into f(x) and simplify, leading us to the conclusion that f(g(x)) = x.
• Similarly, reversing the order and evaluating g(f(x)), by substituting f(x) into g(x), should also yield the same result: g(f(x)) = x.

By showing both conditions hold, we authenticate that f(x) and g(x) are indeed inverses analytically—potentially the most definitive proof of their inverse relationship.

Graphical Verification of Inverse Functions

Beyond the realms of numbers and formulas, we can validate the inverse relationship visually. Graphing both functions on the same coordinate plane allows us to examine their symmetry relative to the line y = x.
Here's how we can approach this graphical verification:

• Plot the function f(x) and its proposed inverse g(x) on a graph.
• Draw the line y = x. This line acts as a mirror, reflecting points over to their inverse positions.
• Observe if every point on the graph of f(x) has a corresponding point on g(x) that is mirrored over the line y = x. If this is true for all points, the functions are inverses graphically.

In our example, the function e^{2x-1} and its inverse \(\frac{1}{2}+\text{ln} \text{\textbackslash}sqrt{x}\) when plotted, should reflect over this line, revealing their inverse nature to our eyes.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are a pair of inverse functions that have a multitude of applications across various scientific fields. Here's a brief explanation:

• An exponential function is of the form f(x) = b^x, where b is a positive constant. It describes rapid growth or decay, such as population growth or radioactive decay.
• A logarithmic function, given by g(x) = log_b(x), inverts this process. It determines the exponent needed to raise the base b to obtain the value x.

The natural base e is significant in the realm of continuous growth, leading to the special exponential function e^x and its inverse, the natural logarithm ln(x). Understanding their relationship helps clarify why, for instance, the function e^{2x-1} is inverted by \(\frac{1}{2}+\text{ln} \text{\textbackslash}sqrt{x}\) as seen in our example.

Symmetry Along the Line y=x

The concept of symmetry is especially powerful in mathematics, providing a visual way to recognize inverse functions. The line y = x serves as a perfect mirror for this purpose.
For any function f(x) and its inverse g(x), symmetry along y = x means:

• For a point (a, b) on the curve of f(x), there will be a corresponding point (b, a) on the curve of g(x).
• The curves of f(x) and g(x) will intersect the line y = x at their points of reflection, reinforcing their inverse relationship.

In our exercise, when we graph both functions f(x) = e^{2x - 1} and g(x) = \(\frac{1}{2} + \text{ln} \text{\textbackslash}sqrt{x}\), their reflection across y = x confirms what we already deduced analytically—f(x) and g(x) are indeed inverse functions.