Question

In Exercises, use a graphing utility to verify that the functions are equivalent for \(x>0\). $$ \begin{aligned} &f(x)=\ln \sqrt{x\left(x^{2}+1\right)} \\ &g(x)=\frac{1}{2}\left[\ln x+\ln \left(x^{2}+1\right)\right] \end{aligned} $$

Short answer

The functions \(f(x) = \ln \sqrt{x\left(x^{2}+1\right)}\) and \(g(x) = \frac{1}{2}\left[\ln(x) + \ln(x^{2}+1)\right]\) are equivalent for \(x > 0\). The logarithmic properties make it easier to demonstrate their equivalence.

Step-by-step solution

Identify the Logarithmic property

The logarithmic property that will be used is \(\ln(ab) = \ln{a} + \ln{b}\). This property helps to simplify expressions and particularly aid in the process of checking equivalence when a product or quotient is involved. In this case, \(\ln \sqrt{x\left(x^{2}+1\right)}\) can be described as \(\ln(a)+\ln(b)\).

Apply the Logarithmic Property

Apply the property to the function \(f(x)\), \(\ln \sqrt{x\left(x^{2}+1\right)} = \frac{1}{2} (\ln(x) + \ln(x^{2}+1))\). The square root can be translated into a power of 1/2, and the resulting expression matches the format \(\ln(a)+\ln(b)\). Also note that \(1/2\) into both terms within the brackets on the right side.

Verify the equivalence

The expression is now the same as the one in the function \(g(x)\). Hence the functions are equivalent for any \(x>0\). You could use a graphing utility to graph both \(f(x)\) and \(g(x)\) to visually confirm that they are indeed the same.

Key concepts

Logarithmic Properties

Logarithmic properties are crucial in simplifying mathematical expressions and understanding the relationship between different functions. One important property is \( \ln(ab) = \ln{a} + \ln{b} \). This property states that the logarithm of a product is equal to the sum of the logarithms of the factors. It is especially useful when manipulating complex expressions involving multiplication.
This property helps break down seemingly complicated functions into more manageable parts, allowing us to perform further algebraic manipulation. For instance, consider the expression \( \ln\sqrt{x(x^2+1)} \). Here, the square root can be expressed as a power of 1/2, leading to \( \ln{[x(x^2+1)]^{1/2}} \)  —  which simplifies with our logarithmic property to \( \frac{1}{2}\left(\ln{x} + \ln{(x^2+1)}\right) \).
This step forms the basis of understanding that the two given functions \( f(x) \) and \( g(x) \) in the exercise are equivalent.

Graphing Utility

Graphing utilities are wonderful tools when tackling equivalences in functions. They enable us to visualize how functions behave across different values of \( x \). By graphing both functions \( f(x) \) and \( g(x) \) using a graphing calculator or software, you can see if the plots are identical. If they overlap perfectly, it further confirms the equivalency that we have determined algebraically.

• Start by inputting the function equations into the graphing utility.
• Pay attention to the domain specified, which in this case is \( x > 0 \).
• Examine the graphs carefully, checking for overlaps and comparing characteristics like intercepts and slopes.
• Confirm that both functions follow the same path, covering the same points for all positive \( x \)."

If done correctly, you will observe that both graphs sit exactly on top of each other, implying that the functions are indeed equivalent visually.

Function Verification

Function verification is the process of establishing that two functions are identical by comparing their algebraic expressions and confirming their equivalence graphically or numerically. With the functions given, algebraic verification involves the use of logarithmic properties to manipulate and simplify the expressions until they appear the same.
To verify the functions satisfying \( x > 0 \), you may;

• Apply algebraic manipulation using logarithmic properties to show similarity in the expressions.
• Use the outcome of logarithmic manipulation to ensure the expressions of \( f(x) \) and \( g(x) \) are exactly the same.
• Graph both functions to visually confirm the algebraic equivalence.

The combination of algebraic simplification and graphical representation provides a comprehensive verification method. These steps ensure students thoroughly understand and can independently verify equivalence between functions even when the expressions are complex.