Problem 54 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=6-\frac{3}{4}(2 x-4) $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = 6 - \frac{2}{3}x \).

Step 1: Simplify the Original Function

The given function is \( f(x) = 6 - \frac{3}{4}(2x - 4) \). Simplify the function by distributing the \( \frac{3}{4} \) across \( (2x - 4) \): \[ f(x) = 6 - \frac{3}{4}\cdot 2x + \frac{3}{4}\cdot 4 \] which simplifies to \( f(x) = 6 - \frac{3}{2}x + 3 \). Combine the constant terms: \[ f(x) = 9 - \frac{3}{2}x \].

Step 2: Set f(x) Equal to y

To find the inverse function, first express the function as \( y = 9 - \frac{3}{2}x \).

Step 3: Solve for x in terms of y

Rearrange the equation \( y = 9 - \frac{3}{2}x \) to solve for \( x \). Start by isolating the term with \( x \): \( y = 9 - \frac{3}{2}x \Rightarrow \frac{3}{2}x = 9 - y \). Now, solve for \( x \): \[ x = \frac{2}{3}(9 - y) \], which can be rewritten as \[ x = 6 - \frac{2}{3}y \].

Step 4: Write the Inverse Function

To express \( f^{-1}(x) \), swap \( x \) and \( y \) in \( x = 6 - \frac{2}{3}y \), which yields \( y = 6 - \frac{2}{3}x \). Thus, the inverse function is: \[ f^{-1}(x) = 6 - \frac{2}{3}x \].

Key Concepts

Symbolic Representation

In mathematics, finding the symbolic representation of inverse functions is like turning a function inside out. When dealing with inverse functions, we're striving to express a function in such a way that if the original function takes a value and gives a result, the inverse takes that result and gives the initial value back. This is symbolically represented as \( f^{-1}(x) \), meaning it's the inverse of \( f(x) \).

In our exercise, we aimed to find a symbolic representation of \( f^{-1}(x) \) where the function was initially given as \( f(x) = 6 - \frac{3}{4}(2x - 4) \). The process involves manipulating the function to express it in terms of \( y \), solving for \( x \), and ultimately rewriting it to express \( y \) in terms of \( x \). This process can initially be confusing, but through systematic symbolic manipulation and algebraic steps, we arrive at the desired inverse function.

Function Simplification

Function simplification is a crucial step when working with functions. It involves reducing a given function to its simplest form, making it easier to interpret and work with. In the exercise, the function was given as \( f(x) = 6 - \frac{3}{4}(2x - 4) \).

To simplify this, we distributed the fraction \( \frac{3}{4} \) across the expression \((2x - 4)\) to get \( 6 - \frac{3}{2}x + 3 \). This expression simplifies further to \( 9 - \frac{3}{2}x \) by combining like terms — the constants 6 and 3.

Function simplification not only helps in understanding but also makes further calculations, such as finding the inverse function, much more manageable. The simplification process can turn initially complicated expressions into a form that reveals the basic properties of the function more clearly.

Algebraic Manipulation

Algebraic manipulation is all about using algebraic rules and operations to transform a mathematical expression or equation. This technique is essential when finding an inverse function, as it requires us to solve for one variable in terms of another.

In the problem, we converted the simplified function \( y = 9 - \frac{3}{2}x \) by performing algebraic steps to solve for \( x \) in terms of \( y \). This included adding and subtracting values appropriately and dividing through by coefficients attached to the variable. For instance, \( \frac{3}{2}x = 9 - y \) simplifies to \( x = \frac{2}{3}(9 - y) \), or equivalently, \( x = 6 - \frac{2}{3}y \).

Through careful algebraic manipulation, we derived the inverse and, after swapping \( x \) and \( y \), finalized the inverse function as \( f^{-1}(x) = 6 - \frac{2}{3}x \). Mastering these algebraic skills is essential for solving various mathematical problems efficiently.