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Magazine Chapter 2: Problem 64
Find the inverse function of \(f.\) $$f(x)=\frac{1}{x^{2}}, \quad x>0$$
Short Answer
Expert verified
The inverse function is \( f^{-1}(x) = \frac{1}{\sqrt{x}} \).
Step by step solution
01
Understand the Given Function
The function provided is \( f(x) = \frac{1}{x^2} \) for \( x > 0 \). It means we first need to find a function that, when composed with \( f \), results in the identity function. We denote the inverse function as \( f^{-1}(y) \).
02
Set Up the Equation
To find the inverse, start by replacing \( f(x) \) with \( y \): \( y = \frac{1}{x^2} \). This equation relates \( y \) to \( x \) through the function \( f \). Our goal is to solve for \( x \) in terms of \( y \).
03
Solve for x
Rearrange the equation \( y = \frac{1}{x^2} \) to solve for \( x \). Multiply both sides by \( x^2 \) to get \( yx^2 = 1 \). Then, divide both sides by \( y \) to get \( x^2 = \frac{1}{y} \).
04
Solve for x in Terms of y
Take the square root of both sides to solve for \( x \). Since \( x > 0 \), we consider only the positive root: \( x = \sqrt{\frac{1}{y}} = \frac{1}{\sqrt{y}} \). Thus, the inverse function is \( f^{-1}(y) = \frac{1}{\sqrt{y}} \).
05
Swap Variables for Final Expression
Since we usually express inverse functions in terms of \( x \), replace \( y \) with \( x \) in the inverse function. Thus, \( f^{-1}(x) = \frac{1}{\sqrt{x}} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Finding the Inverse
To understand inverse functions, imagine a function that reverses the effect of the original function. The purpose of finding the inverse is to obtain a new function that, when applied, yields the original input value. We denote this new function as \( f^{-1}(x) \). For a straightforward step-by-step approach:
- First, replace the function's output \( f(x) \) with \( y \) in the equation.
- Next, solve this equation for \( x \) in terms of \( y \).
- The resulting equation is the inverse, but it's formulated using \( y \).
- Finally, swap the variables for a more common representation, changing \( y \) back to \( x \), so you have \( f^{-1}(x) \).
Solving Equations
An essential aspect of finding an inverse function is solving equations efficiently. It involves algebraically manipulating the function equation to isolate a variable. Here's how you can solve equations to find the inverse:1. **Reversing Operations**: Take the function, and start by undoing operations like reciprocal, square, etc.Suppose you have \( y = \frac{1}{x^2} \). Here, follow these steps:
- Multiply both sides by \( x^2 \) to remove the fraction, resulting in \( yx^2 = 1 \).
- Divide both sides by \( y \) to isolate \( x^2 \), so you have \( x^2 = \frac{1}{y} \).
- Apply the square root to both sides, considering only positive roots for \( x > 0 \), to achieve \( x = \frac{1}{\sqrt{y}} \).
Functions and Graphs
Functions represent relationships between sets of numbers, usually inputs and outputs. Understanding how they relate to their inverses on graphs enriches comprehension.To visualize, consider:- **A Function Graph**: It often shows the behavior of a function, such as \( f(x) = \frac{1}{x^2} \) being a decreasing function for \( x > 0 \).- **An Inverse on Graphs**: The inverse function \( f^{-1}(x) \) usually reflects over the line \( y = x \).With the example \( f(x) = \frac{1}{x^2} \),
- If you plot it, you'll see a curve that never reaches the axis, always staying in the positive quadrant.
- When plotting its inverse \( f^{-1}(x) = \frac{1}{\sqrt{x}} \), the curve flips such that each point on this graph corresponds to its original counterpart mirrored over the line \( y = x \).
