Problem 47 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=\frac{2 x+5}{x-7} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{7x + 5}{x - 2} \).

Step 1: Replace f(x) with y

Let's start by replacing \( f(x) \) with \( y \) in the given equation to make it easier to work with. So, the equation becomes: \( y = \frac{2x+5}{x-7} \).

Step 2: Swap x and y

To find the inverse, we interchange \( x \) and \( y \). This gives us: \( x = \frac{2y+5}{y-7} \).

Step 3: Solve for y

Our task is to solve for \( y \). Start by multiplying both sides by \( y-7 \) to eliminate the fraction:\[ x(y-7) = 2y+5 \].

Step 4: Expand and Rearrange the Equation

Expand the equation: \( xy - 7x = 2y + 5 \). Rearrange to group \( y \) terms on one side: \( xy - 2y = 7x + 5 \).

Step 5: Factor out y

Factor out \( y \) from the left side equation: \( y(x - 2) = 7x + 5 \).

Step 6: Solve for y

Now, divide both sides by \( x - 2 \):\[ y = \frac{7x + 5}{x - 2} \].

Step 7: Write the inverse function

Thus, the inverse function of \( f \) is \( f^{-1}(x) = \frac{7x + 5}{x - 2} \).

Key Concepts

Understanding Function Transformation

Function transformation involves changing a function's form to achieve specific manipulations or observations. In the context of finding an inverse function, transformation is crucial because the roles of inputs and outputs (usually denoted as variables in functions) are switched.

Here's what transformation looks like in the process of finding an inverse:

• Initially, you have a function, like our given function \( f(x) = \frac{2x+5}{x-7} \), where the output depends on the expression \(2x+5\) divided by \(x-7\).
• To transform this into its inverse, swap the positions of \( x \) and \( y \) in your function equation, so each input becomes an output and vice versa.
• This swap reflects a central aspect of inverse function transformation, where every new \( y \) value corresponds to an original \( x \) value and vice versa.

This transformation helps us reconstruct the function from its output form back to its original input form. The exchange of \( x \) and \( y \) is a pivotal manipulation setting the stage for solving the inverse equation, allowing for insights into how transformations inform inverse calculations.

Algebraic Manipulation Techniques

Algebraic manipulation is critical in solving equations and reshaping them into a more workable form. When finding an inverse function, this skill enables you to rearrange, isolate, and solve variables effectively.

Consider how algebraic manipulation aids in our exercise:

• After swapping \( x \) and \( y \), your task is to isolate \( y \) to find its expression in terms of \( x \). Start by clearing fractions, a common algebraic step, which involves multiplying both sides of the equation by the denominator to facilitate further operations.
• Next, expansion and reorganization of terms helps in isolating \( y \). It involves distributing multiplication over any terms and rearranging similar terms to the same side.
• Factoring helps in isolating \( y \). By pulling out \( y \) as a common factor, you can simplify expressions, making division only necessary for solving \( y \).

These manipulations reflect key algebraic skills. They enable the transformation of an equation from one form into another and systematically guide you to the inverse function. Mastering these skills allows you to solve complex problems by breaking them into simpler, manageable steps.

Diving Into Rational Functions

Rational functions are fractions involving polynomial expressions, composed of numerators and denominators. Understanding their behavior is essential, especially when working with inverse functions like in our exercise.

Key characteristics of rational functions involve:

• The function's form \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. For example, our function \( \frac{2x + 5}{x - 7} \).
• Important to inverse functions is identifying domains and potential restrictions or asymptotes. For instance, your original function is undefined where its denominator \( x - 7 = 0 \), thus \( x eq 7 \).
• The inverse function you obtained is also a rational function: \( f^{-1}(x) = \frac{7x + 5}{x - 2} \), needing similar attention to its own restrictions, such as \( x eq 2 \) to avoid division by zero.

Understanding these functions and their characteristics helps in recognizing domain restrictions and simplifying operations needed to invert the function correctly. They provide a robust framework for dealing with real-world scenarios where such inverses may apply, like in engineering and physics problems.