Question

Manipulating Functions.If \(y=5 x+3,\) write \(x=f(y)\).

Short answer

To express x as a function of y given that y=5x+3, solve for x resulting in the function x=f(y) = (y - 3)/5.

Step-by-step solution

Understand the Problem

The given function is in the form of y as a function of x, expressed as y=5x+3. The goal is to express x as a function of y, which will be written as x=f(y).

Isolate the Variable x

To isolate x, we need to solve the given equation for x. Start by subtracting 3 from both sides of the equation to get y - 3 = 5x.

Divide by the Coefficient of x

Next, divide everything by the coefficient of x, which is 5, to get (y - 3)/5 = x. This gives us x as a function of y.

Write x as f(y)

Now that x has been isolated, rewrite it as x=f(y), where f(y) represents the expression we found. Thus, x=f(y) = (y - 3)/5.

Key concepts

Understanding Inverse Functions

When dealing with inverse functions, it is essential to understand that an inverse function reverses the effect of the original function. If you have a function that takes an input 'x' and produces an output 'y', the inverse function takes 'y' as its input and gives you back 'x'.

The relationship between a function and its inverse is akin to a two-way street: if the original function represents the trip from home to work, the inverse function represents the journey back from work to home. To find the inverse, we perform operations that will undo the effect of the original function. In our example exercise where the function is given by the equation \(y=5x+3\), finding the inverse essentially means figuring out how to start with 'y' and wind up at 'x'.

Steps to Finding an Inverse

• First, swap 'x' and 'y' in the equation. So, our equation becomes \(x=5y+3\).
• Second, solve for 'y' as you would solve any linear equation—to isolate 'y' on one side of the equation.
• After applying these steps, your new equation represents the inverse function, which often is denoted as \(f^{-1}(x)\).

Keep in mind, not all functions have inverses that are also functions; for a function to have an inverse that is also a function, it must pass the horizontal line test.

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are straightforward because they can be solved in a finite number of steps and have at most one solution.

When we approach a linear equation, our end goal is to isolate the variable—get it by itself on one side of the equation, with a coefficient of 1. The process usually involves applying inverse operations, such as adding or subtracting the same number from both sides or multiplying or dividing by the same number. It’s like untangling a piece of string: each move you make aims to free the variable from the numbers around it.

Key Points for Solving Linear Equations:

• Perform the same operation on both sides of the equation to maintain the balance.
• Be systematic; deal with addition/subtraction first and then multiplication/division.
• Check your solution by substituting it back into the original equation.

This method is not only used for finding the solution to an equation but also to manipulate equations to change the form of the function, like finding an inverse.

Isolating Variables

Isolating variables is a critical process in algebra whereby we intend to 'get the variable alone' on one side of an equation. This process allows us to find what the variable represents in terms of the other numbers or variables in the equation.

The ability to isolate a variable is the essence of solving equations and forms the basis of understanding more complex algebraic expressions. To isolate a variable involves a series of steps that typically include addition, subtraction, multiplication, and division on both sides of the equation, always aiming to simplify and get the variable by itself.

Effective Strategies for Isolation

• Simplify both sides of the equation if needed, combining like terms and simplifying fractions.
• Undo addition or subtraction first by doing the inverse operation.
• Eliminate multiplication or division once the variable is only part of a product or quotient.
• Keep the equation balanced by doing the same thing to both sides.
• Double-check by replacing the isolated variable back into the original equation.

An isolated variable is the goal when manipulating the function to write it in an inverse form or when solving for an unknown in algebra.