Problem 16
Question
For the following exercises, find the inverse of the functions. $$ f(x)=\sqrt{2 x+1} $$
Step-by-Step Solution
Verified Answer
The inverse of \( f(x) = \sqrt{2x+1} \) is \( f^{-1}(x) = \frac{x^2 - 1}{2} \).
1Step 1: Identify the Function
We are given the function \( f(x) = \sqrt{2x+1} \). Our goal is to find its inverse function, \( f^{-1}(x) \).
2Step 2: Set the Function Equal to y
Start by letting \( y = f(x) = \sqrt{2x+1} \). This helps to work with it algebraically.
3Step 3: Swap Variables
To find the inverse, swap the roles of \( x \) and \( y \). So, rewrite the equation as \( x = \sqrt{2y+1} \).
4Step 4: Isolate the Square Root
Square both sides to eliminate the square root: \( x^2 = 2y+1 \). This simplifies our equation and allows us to isolate \( y \).
5Step 5: Solve for y
Rearrange the equation to solve for \( y \): \( 2y = x^2 - 1 \), which results in \( y = \frac{x^2 - 1}{2} \).
6Step 6: Write the Inverse Function
The inverse function of \( f(x) \) is \( f^{-1}(x) = \frac{x^2 - 1}{2} \).
Key Concepts
Understanding Function NotationThe Process of Solving EquationsDemystifying the Square Root Function
Understanding Function Notation
Function notation is a way to write mathematical functions using symbols to convey a clear and accurate representation. It is typically denoted as \( f(x) \), where \( f \) represents the function name, and \( x \) represents the input variable.
This notation offers a concise and efficient way of expressing how a function transforms an input into an output.
When we have an equation like \( f(x) = \sqrt{2x + 1} \), it tells us that for every value of \( x \), the function \( f \) transforms it into the square root of \( 2x + 1 \).
Why is function notation important?
This notation offers a concise and efficient way of expressing how a function transforms an input into an output.
When we have an equation like \( f(x) = \sqrt{2x + 1} \), it tells us that for every value of \( x \), the function \( f \) transforms it into the square root of \( 2x + 1 \).
Why is function notation important?
- It helps in identifying the input and output clearly.
- Facilitates easier operations on the function, like finding inverses.
- Makes it universal and understandable across different contexts.
The Process of Solving Equations
Solving equations involves finding the values of variables that make an equation true. It's a critical skill in mathematics, as it allows us to unravel complex problems.
In our example, to find the inverse function, we engage in solving equations. The goal is to express the new variable \( y \) in terms of \( x \). Here's a quick breakdown of solving:
This will guide you towards finding the exact solution needed, such as the inverse function \( f^{-1}(x) = \frac{x^2 - 1}{2} \).
In our example, to find the inverse function, we engage in solving equations. The goal is to express the new variable \( y \) in terms of \( x \). Here's a quick breakdown of solving:
- Identify the equation to work with, like swapping variables to get \( x = \sqrt{2y + 1} \).
- Perform mathematical operations to isolate the variable of interest, which involves "undoing" operations. For instance, squaring both sides gets rid of the square root, leading to \( x^2 = 2y + 1 \).
- Rearrange the equations as needed to solve for the desired variable. This can involve operations like subtraction, division, etc.
This will guide you towards finding the exact solution needed, such as the inverse function \( f^{-1}(x) = \frac{x^2 - 1}{2} \).
Demystifying the Square Root Function
The square root function is an essential mathematical concept characterized by the operation of extracting a square root, typically written as \( \sqrt{x} \).
In the context of our function \( f(x) = \sqrt{2x + 1} \), it tells us that for each input \( x \), the function calculates the square root of \( 2x + 1 \). The square root function helps in defining the output calculation in inverse operations and transformations.
Key points about the square root function:
In the context of our function \( f(x) = \sqrt{2x + 1} \), it tells us that for each input \( x \), the function calculates the square root of \( 2x + 1 \). The square root function helps in defining the output calculation in inverse operations and transformations.
Key points about the square root function:
- Always produces non-negative results since the root of a negative number isn’t considered in real numbers.
- Illustrates a very specific constraint: the expression under the root must be zero or positive. Hence, \( 2x + 1 \) must be \( \geq 0 \) for real solutions.
- Square rooting is reversible, which is crucial in finding inverse functions, as seen in our swap to \( x = \sqrt{2y + 1} \).
Other exercises in this chapter
Problem 15
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=
View solution Problem 16
For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly with \(x\) and \(z\) and when \(x=2\) an
View solution Problem 16
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. $$ f(x)=\frac{x^{2}-1}{x^{3}+9 x^{2}+14 x} $$
View solution Problem 16
For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. $$ f(x)=3 x^{3}+x^{2}-20 x+12 ; \qu
View solution