Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions. $$ f(x)=\sqrt{3-4 x} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{3 - x^2}{4} \).

Step 1: Express the function with y

Set the function equal to y: \[ y = \sqrt{3 - 4x} \] This expresses the function in terms of y instead of f(x), which will make it easier to manipulate algebraically.

Step 2: Swap x and y

To find the inverse function, swap x and y: \[ x = \sqrt{3 - 4y} \] The goal is to solve for y in terms of x, as the inverse function will be denoted as \( f^{-1}(x) \).

Step 3: Square both sides to eliminate the square root

Square both sides of the equation to get rid of the square root:\[ x^2 = 3 - 4y \] This step simplifies the equation by removing the square root, making it easier to solve for y.

Step 4: Isolate the term containing y

Rearrange the equation to isolate the term with y:\[ 4y = 3 - x^2 \] Here, we subtract \( x^2 \) from 3 to keep the y-term alone on one side of the equation.

Step 5: Solve for y

Divide both sides of the equation by 4 to solve for y:\[ y = \frac{3 - x^2}{4} \]This provides the expression for y in terms of x, which represents the inverse of the original function.

Step 6: Write the inverse function

Now that we've solved for y, write the inverse function:\[ f^{-1}(x) = \frac{3 - x^2}{4} \]This function is the inverse of the original function \( f(x) = \sqrt{3 - 4x} \).

Key Concepts

Function Manipulation

When working with functions, manipulating them is essential for various purposes like finding inverses. In our exercise, the original function given is \( f(x) = \sqrt{3 - 4x} \). Understanding how to manipulate functions is crucial because it allows us to find different representations or solve specific problems. One type of manipulation is replacing \( f(x) \) with \( y \), which helps simplify our operations to find the inverse.
To find the inverse, key manipulations include swapping variables and rearranging terms. Swapping \( x \) and \( y \) helps target the inverse function. We change the focus from \( y \) depending on \( x \) to \( x \) depending on \( y \). This step is critical: it rewrites the function's relationship, setting the stage for solving for the new \( y \).

• Switching variables: Swap \( x \) and \( y \)
• Simplifying expressions: Rearrange equations to isolate terms

These manipulations are reversible and aim to reflect the inherent symmetry between a function and its inverse. Successfully applying these techniques unlocks the door to solving equations within the context of any function.

Algebraic Expressions

Algebra is powerful when dealing with expressions and equations in functions. The problem presents us with \( y = \sqrt{3 - 4x} \), which is our primary algebraic expression.
To discover the inverse, transforming this expression into other forms is vital. Observations about the operations involved, such as square roots and constants, are the first step. Handling square roots particularly needs care. During manipulation, such as in inverses, it's necessary to eliminate the square root for full isolation of variables.

• Square both sides: \( x^2 = 3 - 4y \). This action simplifies the expression, allowing further action in isolating \( y \).
• Rearrange terms: Moving terms around, such as subtracting or adding from both sides, is a basic algebraic maneuver to reach the desired form.

Understanding the persistence of algebraic identities like \( a^2 + b^2 \) in transformations is beneficial. Here, algebra simplifies our inverse, sculpting it into the neat expression \( f^{-1}(x) = \frac{3 - x^2}{4} \), which is elegant and insightful.

Solving Equations

Solving equations is the heart of finding inverse functions. It's about unraveling one variable in terms of another. In this exercise, we start with \( x = \sqrt{3 - 4y} \).
Our aim is to solve for \( y \), entailing several algebraic steps:

• Isolating variables: The step-by-step re-arrangement until terms containing \( y \) are on one side.
• Inverting operations: Actions like squaring transform expressions, allowing complex functions to be solved.

The progression of solving the equation involves several algebraic tools: squaring, simplifying terms, and dividing by constants. When the equation reaches the form \( 4y = 3 - x^2 \), we see our steps converging towards the answer. This clarity in isolating \( y \) entails dividing by the coefficient of \( y \) itself: \( y = \frac{3 - x^2}{4} \).
Ultimately, solving this final expression provides the inverse function. Clear steps in solving foster a better understanding. With each tidy step, equations become less about numbers and more about understanding their behavior and relationships, especially as they describe inverse functions.