Problem 27 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ f(x)=\frac{7 x-4}{8} $$

Step-by-Step Solution

Answer

The inverse of \( f(x) \) is \( f^{-1}(x) = \frac{8x+4}{7} \).

Step 1: Replace f(x) with y

The given function is \( f(x) = \frac{7x-4}{8} \). First, replace \( f(x) \) with \( y \). Thus, we have:\[y = \frac{7x-4}{8}\]

Step 2: Interchange x and y

To find the inverse of a function, switch the roles of \( x \) and \( y \) in the equation. So the equation now becomes:\[x = \frac{7y-4}{8}\]

Step 3: Solve for y

Now solve the equation for \( y \) to find the inverse function. Start by multiplying both sides by 8 to eliminate the fraction:\[8x = 7y - 4\]Next, add 4 to both sides:\[8x + 4 = 7y\]Finally, divide both sides by 7:\[y = \frac{8x+4}{7}\]So, the inverse function is \( f^{-1}(x) = \frac{8x+4}{7} \).

Step 4: Graph the function and its inverse

To visualize both the function and its inverse, plot \( f(x) = \frac{7x-4}{8} \) along with its inverse \( f^{-1}(x) = \frac{8x+4}{7} \) on the same coordinate plane. The graphs will be reflections of each other over the line \( y = x \). Additionally, this line can be drawn to better illustrate the reflection.

Key Concepts

Function Transformation

Function transformation is the process of modifying the characteristics of a function without altering its fundamental nature. In the context of inverse functions, a key transformation involves reversing the operations performed by the original function. This allows us to find a function that "undoes" the work of the original.Here’s what you need to know:

• When finding an inverse, you first swap the input and output variables, typically represented by switching the roles of "x" and "y."
• The transformation involves operations such as reversing addition and subtraction, as well as multiplicative operations.
• This process of inversion changes how the function behaves, essentially "mirroring" it across the line \( y = x \).

Understanding function transformation helps you see the symmetry and balance in functions and their inverses, as they retrace each other's steps.

Graphing Functions

Graphing functions and their inverses is an invaluable tool in visualizing relationships. It provides a clear picture of how these functions behave over their domains.To graph the function \( f(x) = \frac{7x - 4}{8} \) and its inverse \( f^{-1}(x) = \frac{8x + 4}{7} \):

• Start by calculating several input-output pairs for both functions. Substitute different values of \( x \) into both \( f(x) \) and \( f^{-1}(x) \).
• Plot these points on a coordinate grid. You’ll notice that corresponding points between the function and its inverse will be reflections over the line \( y = x \).
• Draw the line \( y = x \). This line acts as the mirror line, helping you see the reflective symmetry between the original function and its inverse.

Graphing aids intuition, helping dissolve abstract concepts into tangible patterns that showcase the profound concept of inverse functions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and solving equations to find desired variables. This is crucial when finding inverse functions, as it involves solving for "y" in terms of "x."Here are the steps in algebraic manipulation for finding an inverse function:

• Rewrite the function using "y" for \( f(x) \). This sets up the equation to be manipulated.
• Swap "x" and "y." This swap reflects the idea of inputs becoming outputs, an essential characteristic of inverses.
• Perform algebraic operations to isolate "y." This may include adding, subtracting, multiplying, or dividing to solve for "y." These operations "undo" the effects of the function's operations.
• Once isolated, express "y" as a new function of "x." This expression represents the inverse function.

Algebraic manipulation is the procedural backbone that transforms equations, enabling us to unlock inverse relationships with precision.