Problem 26 - Contemporary Precalculus

Question

Use the Round-Trip Theorem on page 223 to show that \(g\) is the inverse of \(f\) $$f(x)=\frac{-3}{2 x+5}, \quad g(x)=\frac{-3-5 x}{2 x}$$

Step-by-Step Solution

Answer

Question: Show that g is the inverse of f using the Round-Trip Theorem, where f(x) = (-3) / (2x+5) and g(x) = (-3-5x) / (2x). Answer: Since we have proven that both f(g(x)) = x and g(f(x)) = x, it confirms that g is indeed the inverse of f, according to the Round-Trip Theorem.

Step 1: Calculate f(g(x))

To find f(g(x)), we need to substitute g(x) into the expression for f(x): $$ f(g(x)) = f\left(\frac{-3-5x}{2x}\right) = \frac{-3}{2 \left(\frac{-3-5x}{2x}\right) +5} $$ Now, let's simplify this expression.

Step 2: Simplify f(g(x))

We need to simplify the denominator by adding the term inside the parenthesis: $$ \begin{aligned} f(g(x)) &= \frac{-3}{2 \left(\frac{-3-5x}{2x}\right) +5}\\ &= \frac{-3}{\frac{-6-10x}{2x}+5} \end{aligned} $$ Now, let's find a common denominator and combine the terms in the denominator: $$ \begin{aligned} f(g(x)) &= \frac{-3}{\frac{-6-10x+10x}{2x}}\\ &= \frac{-3}{\frac{-6}{2x}} \end{aligned} $$ Finally, let's simplify the fraction: $$ \begin{aligned} f(g(x)) &= \frac{-3}{\frac{-6}{2x}}\\ &= \frac{-3}{-3} \cdot \frac{2x}{1}\\ &= x \end{aligned} $$ Now that we have found f(g(x)) = x, we need to find g(f(x)).

Step 3: Calculate g(f(x))

To find g(f(x)), we need to substitute f(x) into the expression for g(x): $$ g(f(x)) = g\left(\frac{-3}{2x+5}\right) = \frac{-3-5\left(\frac{-3}{2x+5}\right)}{2\left(\frac{-3}{2x+5}\right)} $$ Now, let's simplify this expression.

Step 4: Simplify g(f(x))

We need to simplify the numerator and the denominator: $$ \begin{aligned} g(f(x)) &= \frac{-3-5\left(\frac{-3}{2x+5}\right)}{2\left(\frac{-3}{2x+5}\right)}\\ &= \frac{-3+\frac{15}{2x+5}}{\frac{-6}{2x+5}} \end{aligned} $$ Now, let's find a common denominator and combine the terms in the numerator: $$ \begin{aligned} g(f(x)) &= \frac{\frac{-3(2x+5)+15}{2x+5}}{\frac{-6}{2x+5}}\\ &= \frac{-6x-15}{2x+5} \cdot \frac{2x+5}{-6}\\ \end{aligned} $$ Finally, let's simplify the fraction: $$ \begin{aligned} g(f(x)) &= \frac{-6x-15}{2x+5} \cdot \frac{2x+5}{-6}\\ &= \frac{-1}{-1}\cdot \frac{6x+15}{6}\\ &= x \end{aligned} $$

Step 5: Conclusion

Since we have shown that f(g(x)) = x and g(f(x)) = x, we can conclude that g is the inverse of f using the Round-Trip Theorem.

Key Concepts

Round-Trip Theorem

The Round-Trip Theorem is a fundamental principle in understanding inverse functions. This theorem states that if you have two functions, say f and g, and g is the inverse of f, then when you compose f with g (and vice versa), the result should bring you back to where you started, essentially making a 'round trip'. In mathematical terms, this means that f(g(x)) = x and g(f(x)) = x.

Applying this to our example, we calculated f(g(x)) and found it simplifies to x, and then calculated g(f(x)) and also found it simplifies to x. This two-way successful round trip confirms that g and f are indeed inverses of each other according to the Round-Trip Theorem.

Function Composition

Function composition involves combining two functions in a way that the output of one function becomes the input of the other. In the notation f(g(x)), the function g(x) is applied first, and its output is then used as the input for the function f. This creates a new function that can then be simplified or analyzed for various properties.

In the exercise, we first composed f with g by substituting g(x) into f(x), and then we composed g with f in the same way. Through these compositions, we were aiming to demonstrate that each function 'undoes' the effect of the other, which is a key characteristic of inverse functions.

Simplifying Expressions

Simplifying expressions is the process of reducing a complex mathematical expression into a simpler or more comprehensible form without changing its value. The goal is to make the expression easier to understand or work with. This involves combining like terms, finding common denominators, and canceling terms when possible.

In the provided solution, simplification plays an essential role. For both f(g(x)) and g(f(x)), simplifying the complex fractions was required in order to reveal their simplest form, x. Without simplification, it wouldn’t have been as apparent that g is indeed the inverse of f. It's like untangling a knot to reveal a straight line.

Verification of Inverse Functions

Verification of inverse functions is the process of proving that a given function g is the inverse of another function f. This can be accomplished through several methods, but one of the most straightforward is the use of the Round-Trip Theorem.

In our exercise, we've employed this theorem for verification by showing f(g(x)) = x and g(f(x)) = x. By doing this, we have effectively verified that f and g are inverses. It's a powerful confirmation because it doesn't just suggest but rather conclusively proves that g undoes whatever f does, and vice versa. This kind of verification is critical in precalculus because it ensures the functions have the proper relationship to be considered inverses.