Problem 23 - Algebra and Trigonometry

Question

Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. \(f(x)=2 x-5 ; \quad g(x)=\frac{x+5}{2}\)

Step-by-Step Solution

Answer

f(x) and g(x) are inverses as f(g(x)) = x and g(f(x)) = x.

Step 1: Understand Inverse Function Property

Two functions, \(f\) and \(g\), are inverses of each other if \(f(g(x)) = x\) for every \(x\) in the domain of \(g\), and \(g(f(x)) = x\) for every \(x\) in the domain of \(f\). This means applying one function and then the other function will return the original input value.

Step 2: Verify f(g(x)) = x

Start with the expression for \(g(x)\), which is \(g(x) = \frac{x+5}{2}\). Substitute \(g(x)\) into \(f(x)\):\[ f(g(x)) = f\left(\frac{x+5}{2}\right) = 2\left(\frac{x+5}{2}\right) - 5 \] Simplify the expression: \[ = (x+5) - 5 = x \] Thus, \(f(g(x)) = x\), satisfying the inverse condition.

Step 3: Verify g(f(x)) = x

Now take the expression for \(f(x)\), which is \(f(x) = 2x - 5\). Substitute \(f(x)\) into \(g(x)\):\[ g(f(x)) = g(2x - 5) = \frac{(2x - 5) + 5}{2} \] Simplify the expression: \[ = \frac{2x}{2} = x \] Thus, \(g(f(x)) = x\), satisfying the inverse condition.

Step 4: Conclusion

Since both conditions, \(f(g(x)) = x\) and \(g(f(x)) = x\), hold true, the functions \(f(x) = 2x - 5\) and \(g(x) = \frac{x+5}{2}\) are indeed inverses of each other.

Key Concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. In the context of inverse functions, algebra is used to compose and simplify functions to determine their relationships. When we are working with inverse functions, algebraic expressions help us find outputs that correspond back to the original inputs. In essence, using algebra, we are able to test if two functions

• undo each other's operations, and
• return to the original input values.

In the example of functions given, where we have \[ f(x) = 2x - 5 \] and \[ g(x) = \frac{x+5}{2} \], algebra helps us simplify expressions and verify their inverse properties. For instance, substituting one function into another and using operations like multiplication or division is done to check these inverse relationships.

Function Composition

Function composition is a fundamental concept in mathematics and involves combining two functions in a way that the result of one function is used as the input of another. When checking if two functions are inverses, function composition is the key technique used. The exercises involve composing \[ f(g(x)) \] and \[ g(f(x)) \] to verify their identities.Using function composition, we perform operations such as substitution, which means placing one function into another. Here's a breakdown:

• To find \( f(g(x)) \), substitute \( g(x) = \frac{x+5}{2} \) into \( f \'s \) equation: \[ f(g(x)) = 2 \left(\frac{x+5}{2}\right) - 5 \]. This simplifies to \( x \), confirming the inverse property.
• Similarly, to find \( g(f(x)) \), substitute \( f(x) = 2x - 5 \) into \( g \'s \) equation: \[ g(f(x)) = \frac{(2x-5)+5}{2} \]. It also simplifies to \( x \).

Verifying these compositions helps us see that the functions perform opposite operations, leading us back to the original input value, and thus are inverses.

Mathematics Education

Understanding inverse functions is an important aspect of mathematics education as it builds the foundation for solving complex problems later in higher mathematics. The process of exploring and proving that functions are inverses using the Inverse Function Property not only strengthens algebraic skills but also enhances a student’s critical thinking and problem-solving ability. By learning to apply function composition and test equations in this systematic way, students practice:

• Detailed algebraic manipulation, enhancing their attention to mathematical details.
• Logical reasoning, as they need to establish and understand inverse relationships.
• Methodical problem-solving skills, which are crucial for tackling a variety of mathematical tasks.

This educational approach fosters an understanding of how operations work in reverse, setting the stage for further topics such as logarithms, exponents, and trigonometric inverses in higher mathematics coursework.